research papers on fuzzy graphs

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Published: 09 October 2023

On t-intuitionistic fuzzy graphs: a comprehensive analysis and application in poverty reduction

Asima Razzaque 1 ,
Ibtisam Masmali 2 ,
Laila Latif 3 ,
Umer Shuaib 3 ,
Abdul Razaq 4 ,
Ghaliah Alhamzi 5 &
Saima Noor 1

Scientific Reports volume 13 , Article number: 17027 ( 2023 ) Cite this article

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Mathematics and computing

This paper explains the idea of t-intuitionistic fuzzy graphs as a powerful way to analyze and display relationships that are difficult to understand. The article also illustrates the ability of t-intuitionistic fuzzy graphs to establish complex relationships with multiple factors or dimensions of a physical situation under consideration. Moreover, the fundamental set operations of t-intuitionistic fuzzy graphs are proposed. The notions of homomorphism and isomorphism of t-intuitionistic fuzzy graphs are also introduced. Furthermore, the paper highlights a practical application of the proposed technique in the context of poverty reduction within a specific society. By employing t-intuitionistic fuzzy graphs, the research demonstrates the potential to address the multifaceted nature of poverty, considering various contributing factors and their interdependencies. This application showcases the versatility and effectiveness of t-intuitionistic fuzzy graphs as a tool for decision-making and policy planning in complex societal issues.

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Arunodaya Raj Mishra, Pratibha Rani, … Ibrahim M. Hezam

The optimal emergency decision-making method with incomplete probabilistic information

Ming Fu, Lifang Wang, … Haiyan Shao

Applications of complex picture fuzzy soft power aggregation operators in multi-attribute decision making

Tahir Mahmood, Zeeshan Ali & Muhammad Aslam

Introduction

Decision-making is essential to all aspects of existence. This also pertains to organizations. It is one of the most important factors in determining its success or failure. Every manager must make decisions throughout the management cycle, from planning to control. The level of a manager's success is influenced by the efficacy and caliber of his or her decisions. Without being able to make decisions, managers can't do their other jobs, like planning, organizing, supervising, controlling, and staffing. The decision-making process should be cumulative, consultative, and conducive to organizational growth. Fuzzy decision-making environments offer strategies for handling ambiguity and vagueness based on uncertainty. Ambiguity is a type of uncertainty in which it seems possible to choose more than one option from a list of options. It has been shown that fuzzy set theory ( ${\mathbb{FST}}$ ) is a good way to describe situations where the data are not clear or precise. A fuzzy set can handle this by giving each object in a set a certain amount of membership. In reality, however, a person may suppose that an object $"x"$ belongs to a set $A$ to a certain degree, yet not be entirely convinced. In other words, there may be hesitancy or uncertainty about “ x ” degree of participation in $A$ . In ${\mathbb{FST}}$ , there is no way to account for this uncertainty in membership degrees. Zadeh 1 devised a mathematical method called fuzzy set theory ( ${\mathbb{FST}}$ ) to deal with information that comes from computational perception. This information is imprecise, unclear, ambiguous, vague, or doesn't have clear limits. Since its acceptance, this idea has been utilized in numerous technical and scientific domains. The ${\mathbb{FST}}$ has been used successfully in consumer electronics, control systems, image processing, knowledge-based systems, robotics, industrial automation, artificial intelligence, and consumer electronics. This theory has also been used in many areas of operations research, such as project management, decision theory, supply chain management, queue theory, and quality control. Mapari and Naidu 2 studied some properties of ${{\mathbb{FS}}}$ and discussed their application Some introductory texts in this field were written by Kandel 3 , Klir and Yuan 4 , Mendel 5 , and Zimmermann 6 .

The intuitionistic fuzzy set ( ${\mathbb{IFS}}$ ) generalizes the fuzzy set because the indicator function of the ${{\mathbb{FS}}}$ is a particular case of the membership function and non-membership function of the ${\mathbb{IFS}}$ . Atanassov 7 introduced ${\mathbb{IFS}}$ as an extension of Zadeh's idea of fuzzy set, which itself is an extension of the traditional idea of a set. These sets are quite helpful in offering a flexible approach for elaborating the uncertainty and ambiguity inherent in decision making. De et al. 8 presented the ${{\mathbb{IFS}}}$ operations and also demonstrated their various features. Several important aspects of the newly introduced operations on ${{\mathbb{IFS}}}$ were investigated in 9 . The ${{\mathbb{IFS}}}$ is an essential subject in fuzzy mathematics due to its vast range of real-world applications, including pattern recognition, machine learning, decision making, and market forecasting. Ejegwa et al. 10 provided a clear and complete overview of various ${\mathbb{IFS}}$ models in real-world scenarios. Burillo et al. 11 proposed the concept of intuitionistic fuzzy number ( ${{\mathbb{IFN}}}$ ). The ${\mathbb{IFN}}$ is a more general platform for communicating vague, incomplete, or contradictory information while solving multi-criteria decision-making problems and for expressing and reflecting evaluation information across multiple dimensions. Faizi et al. 12 applied the concept of ${{\mathbb{IFS}}}$ in multi criteria group decision making. Dai et al. 13 developed an intuitionistic fuzzy concept-oriented three-way decision model to tackle the ranking and classification problem in intuitionistic fuzzy multi-criteria contexts with the decision-maker's preference. Das et al. 14 suggested a productive method for group MCDM based on intuitionistic multi-fuzzy set theory. These sets have been utilized in MCDM significantly more recently in 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 . In imaging applications, the enhancement of pictures with weak edges presents significant difficulties. Based on ${\mathbb{IFS}}$ , Liu et al. 23 developed a novel image enhancement technique. While color photographs give more information than grayscale images, segmenting color images is a task that is still in progress. The analysis of biomedical images is especially beneficial for numerous purposes. Bouchet et al. 24 presented a method for the segmentation of leukocytes. This method combines the use of RGB color space, ${\mathbb{IFS}}$ , and K-means clustering. Cagman and Karatas 25 introduced the operation and application of intuitionistic fuzzy soft sets ( ${\mathbb{IFSS}}$ ). Ali et al. 26 defined the aggregation operator for complex ${\mathbb{IFSS}}$ and developed their associated properties. Jabir et al. 27 proposed algorithms based on a generalized ${\mathbb{IFSS}}$ and also showed the supremacy of the given methods. Bashir et al. 28 introduced the possibility of ${\mathbb{IFSS}}$ and associated operations. The interval-valued ${\mathbb{IFSS}}$ theory was initiated by Jiang et al. 29 . A definition of a Hausdorff distance-based similarity measure between ${\mathbb{IFSS}}$ and its potential application in medical diagnosis were given in 30 . Deli and Karats 31 established the concept of interval-valued intuitionistic fuzzy parameterized soft sets. They presented a decision-making method based on this notion in 32 . One of the most recent approaches to dealing with imprecision is the Pythagorean fuzzy set (PFS). These sets generalize ${\mathbb{IFS}}$ and have a wider range of uses, which inspires research into their applicability to the problem of career placement. Abdullah et al. 33 depicted the Choquet integral operator based on PFSs. Fuzzy measures can be used to account for how parts of PFSs interact with each other.

A graph is a convenient method for describing data containing object relationships. Relationships are represented by edges and objects by vertices. It is commonly known that graphs are simple representations of relations. Graph theory provides a useful instrument for quantifying and simplifying the numerous moving pieces of dynamic systems. Mathematical chemistry examines the structure of molecules using mathematical methods. Molecular descriptors serve a key role in mathematical chemistry. As a field of study, chemical graph theory shows how chemistry, graph theory, and math are related. A molecular graph is a graph that represents the atoms and bonds of a compound via vertices and edges. With its vertices and edges, the graph makes it easy to see how different things are related to each other. Creating a “Fuzzy Graph Model” may be necessary to clarify the situation if there is any ambiguity in the description of objects or their relationships. They must deal with uncertain situations, and more information requires some high-potential tools. The graph is one such mathematical tool which effectively deals with extensive data. Fuzzy graph is a tool that needs to be used when uncertain factors exist. Rosenfeld 34 took the first step into the field of fuzzy graph. Mordeson and Chang-Shyh 35 discussed certain fuzzy graph operations. Bhattacharya 36 proved several graph theoretic results for fuzzy graph. Bhutani 37 worked on automorphisms of fuzzy graph. The fuzzy graph is used in a wide range of scientific and engineering fields, such as broadcast communications, production, social networks, artificial intelligence, data hypotheses, and neural systems. The study of fuzzy graph led many researchers to contribute in this fields. Pathinathan et al. 38 initiated the idea of hesitant fuzzy graph. Javaid et al. 39 proposed numerous operations on hesitant fuzzy graphs. Moreover, Akram and Saira 40 introduced the notion of fuzzy soft graphs and they also presented the applications of fuzzy soft graphs in social and road networks 41 . Ali et al. 42 initiated the complex q-rung orthopair fuzzy planar graph theory. Kifayat et al. 43 explored the ideas of complex q-rung orthopair fuzzy k-competition, complex q-rung orthopair fuzzy p-competition, and complex q-rung orthopair fuzzy neighborhoods. Fuzzy graphs have been applied to many practical situations like optimization problems 44 , 45 , clustering 46 and social networks 47 . Intuitionistic fuzzy graphs ( ${\mathbb{IFG}}$ ) provide a more accurate representation of human thinking and decision-making processes. Individuals frequently need clarification about the precise acceptance or rejection of an element inside a particular set. The ${\mathbb{IFG}}$ and intuitionistic fuzzy relations were introduced by Shannon and Atanassov 48 and they also looked into some of their characteristics 49 . Karunambigai and Atanassov 50 studied operations on ${\mathbb{IFG}}$ . Gani and Begum 51 discussed the size, order and degree of ${\mathbb{IFG}}$ . Sundas and Akram 52 described the application of an intuitionistic fuzzy soft graph to a problem involving decision-making. Yaqoob et al. 53 developed the complex intuitionistic fuzzy graph theory. Abida and Faryal 54 classified the fundamental operations as direct, semi-strong, strong, and modular products for complex intuitionistic fuzzy graphs. Nandhinii and Amsaveni 55 proposed a bipolar complex intuitionistic fuzzy graph. Furthermore, the literature has extensively examined many principles and applications of ${\mathbb{IFG}}$ and their expansions, as evidenced by the works cited in references 56 , 57 , 58 , 59 . The theory of t- ${\mathbb{IFS}}$ was initiated by Sharma in 60 .

The t- ${\mathbb{IFS}}$ has shown advantages in handling vagueness and uncertainty compared to intuitionistic fuzzy set. It's a good strategy because it gives a flexible way to deal with the uncertainty and ambiguity that come with making decisions. The t-Intuitionistic fuzzy models are becoming more useful because they try to close the gap between traditional numerical models used in engineering and the sciences and symbolic models used in expert systems. The theory of ${\mathbb{IFG}}$ serves as a valuable tool for delineating and clarifying complex and indeterminate matters that arise in practical contexts. This phenomenon can be attributed to its ability to effectively communicate the inherent characteristics of unpredictability, complexity, imprecision, and uncertainty connected with the things encompassed inside these sets. However, it is necessary to rewrite these approaches using specific numerical values to effectively handle the practical concerns related to membership and non-membership functions. To overcome this constraint, we presented the concept of a t- ${\mathbb{IFG}}$ , which utilizes linear t-norm and t-conorm operators. The need for a systematic and adaptable methodology to effectively handle ambiguity and enable decision-making under the guidance of pre-established criteria led to the adoption of the t- ${\mathbb{IFG}}$ . In this context, the utilization of the parameter ‘t’ facilitates the simplification of the procedure by specifying particular criteria for identifying the degree of membership or non-membership. In many practical scenarios, it becomes imperative to make judgments contingent on different levels of confidence. Introducing the parameter ‘t’ in the t- ${\mathbb{IFG}}$ aims to overcome the constraints of the ${\mathbb{IFG}}$ . This parameter offers precise control over stringency, enhances customization, allows for separate thresholds for decision-making, enhances flexibility, and reduces ambiguity. The benefits above render the t- ${\mathbb{IFG}}$ a very effective technique for depicting uncertainty and facilitating well-informed decision-making in contexts that necessitate a tailored and regulated approach to uncertainty management. The t- ${\mathbb{IFG}}$ facilitates the understanding and manipulation of complex decision environments in situations where traditional ${\mathbb{IFG}}$ is insufficient. The role of complicated, ambiguous interactions is essential in the context of decision-making challenges. These graphs thoroughly describe the complex interplay between input and output variables, offering decision-makers powerful tools for analyzing and assessing different choices. Complicated fuzzy connections allow decision-makers to determine options comprehensively and systematically by considering various criteria and their interdependencies. This facilitates a holistic approach to addresses complex decision-making difficulties. The intricate technique represents a significant advancement in decision-making, particularly in situations characterized by membership, non-membership, and parameter t. It signifies a break from the limitations imposed by binary logic and paves the way for enhanced accuracy in decision-making processes.

The subsequent motivation for organizing research is presented:

The primary motivation for using t-IFG is their capacity to effectively handle intricate uncertainty scenarios characterized by hesitant and fluctuating interactions between elements.

By including the "t" parameter, these graphs offer a framework for assessing and modeling diverse levels of uncertainty and confidence in connections.

Incorporating t-norms and t-conorms provides a method for handling the combination and disjunction of uncertain information, designed explicitly for decision-making situations involving a wide range of inputs and outcomes in the real world.

This approach is employed in several fields, like decision analysis, risk assessment, and systems optimization, where the objective is to achieve a trade-off between unknown connections and practical value.

Integrating intuitionistic fuzzy logic, graph theory, and the parameter "t" gives rise to t-intuitionistic fuzzy graphs, which present a novel methodology. The following are the novelties of the present work:

The parameter denoted as "t" represents a threshold that indicates reluctance, enabling the creation of a new and organized representation of unclear connections.

Incorporating the "t" parameter can enhance the depiction of relationships, wherein the selection of edges and nodes is dependent upon keeping to a defined confidence level.

This methodology would provide a more precise differentiation between robust and delicate associations, enabling more systematic handling of ambiguity.

A t- ${\mathbb{IFG}}$ allows for incorporating multi-layered analysis, wherein different graph levels are associated with various parameter values “t”. By employing this approach, it would be possible to thoroughly examine the interconnections within the graph, taking into account different degrees of certainty. It facilitates a deeper understanding of the fundamental framework.

Our primary goals for this article are to make the following contributions:

Propose the idea of the t- ${\mathbb{IFG}}$ . This phenomenon is advantageous in that it offers a flexible paradigm for describing the uncertainty and ambiguity inherent in decision-making. Moreover, it plays a significant role in various disciplines such as computer science, economics, chemistry, medicine, and engineering.

Explore various set theoretical operations of t- ${\mathbb{IFG}}$ and prove many key properties of the newly defined operations. These operations enable the integration of information, the exploration of connections, and the facilitation of informed decision-making across various application domains.

Introduce the notions of homomorphism and isomorphism of t- ${\mathbb{IFG}}$ and demonstrate many newly defined key properties. This notion is used to improve the comfort of conducting comparative analysis and transmitting data in scenarios that include graph topologies that are unsure and hesitant.

Initiate the idea of the complement of a t- ${\mathbb{IFG}}$ and prove many vital properties of this notion. This notion of ambiguity exposes inverse relationships that may not be directly evident in the original graph. The applications of this technology encompass error detection, system verification, and decision analysis.

Identify the critical factors for reducing poverty in a certain society using the newly defined technique. This technique will help reduce poverty by improving representation, identifying susceptible groups, allocating resources, tracking, and evaluating progress, and formulating well-considered policies.

Explores the complexity and uncertainties of poverty, leading to an assessment of the causes, development, and impacts.

Following a brief discussion of the t- ${\mathbb{IFG}}$ , the rest of the paper is structured as follows: In “ Preliminaries ” section, some fundamental definitions are provided to help the reader to comprehend the originality of the work presented in this article. In " t-Intuitionistic Fuzzy Graph " section, the notion of t- ${\mathbb{IFG}}$ is introduced and various fundamental characteristics of this phenomenon are investigated. In " Operations on t-intuitionistic fuzzy graph " section, various set theoretical operations of t- ${\mathbb{IFG}}$ are explored and graphical representations of these operations are demonstrated. In “ Isomorphism of t-intuitionistic fuzzy graphs ” section, the concepts of homomorphisms and isomorphisms of t- ${\mathbb{IFG}}$ are established. In “ Complement of t-intuitionistic fuzzy graph ” section, the idea of complement of t- ${\mathbb{IFG}}$ is defined and many important key features of this notion are explored. In “ Application of t-intuitionistic fuzzy graph ” section, the newly defined strategy is applied to design a mechanism for the reduction of poverty in a certain society. Finally, some comparative analysis and concrete conclusions about the paper are summarized in “ Comparative analysis ” and “ Conclusion ” sections respectively.

The list of abbreviations used in this article is shown in the table below.

Preliminaries

The fundamental concepts and definitions of t- ${\mathbb{IFS}}$ are explained in this section.

Definition 1 7

An ${\mathbb{IFS}}$ $\mathfrak{B}$ of a universe ${\mathbb{U}}$ of the form: $\mathfrak{B}= \{ < {{u}}_{1}, {\mu }_{\mathfrak{B}}\left({{u}}_{1}\right), {\sigma }_{\mathfrak{B}}\left({{u}}_{1}\right)>:{{u}}_{1}\in {\mathbb{U}}\},$ where $\mu_{{\mathfrak{B}}}$ and $\sigma_{{\mathfrak{B}}}$ are the functions from universe ${\mathbb{U}}$ to $\left[ {0, 1} \right],$ respectively, the membership and non-membership of an element $u_{1}$ of the universe ${\mathbb{U}}$ respectively. These functions must satisfy the following condition: $0 \le$ $\mu_{{\mathfrak{B}}} \left( {u_{1} } \right) + \sigma_{{\mathfrak{B}}} \left( {u_{1} } \right) \le 1.$

Definition 2 60

Let ${\mathfrak{B}}$ be an ${\mathbb{IFS}}$ of a universal set ${\mathbb{U}}$ and $t \in \left[ {0,1} \right].$ The ${\mathbb{IFS}}$ ${\mathfrak{B}}_{t}$ of ${\mathbb{U}}$ is called a t-intuitionistic fuzzy set (t- ${\mathbb{IFS}}$ ) and is defined as: $\mu_{{{\mathfrak{B}}_{t} }} \left( {u_{1} } \right) = min\{ \mu_{{\mathfrak{B}}} \left( {u_{1} } \right),t\}$ and $\it \sigma_{{{\mathfrak{B}}_{t} }} \left( {u_{1} } \right) = \max \left\{ {\sigma_{{\mathfrak{B}}} \left( {u_{1} } \right), 1 - t} \right\},\forall u_{1} \in {\mathbb{U}}$ . The value of $\tau \left( {u_{1} } \right) = 1 - \left( {\mu_{{A_{t} }} \left( {u_{1} } \right) + \sigma_{{A_{t} }} \left( {u_{1} } \right)} \right)$ is called the degree of hesitancy. The t- ${\mathbb{IFS}}$ is of the form: ${\mathfrak{B}}_{t} = \left\{ {\left( {u_{1} ,\mu_{{{\mathfrak{B}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathfrak{B}}_{t} }} \left( {u_{1} } \right)} \right):u_{1} \in {\mathbb{U}}} \right\},$ where $\mu_{{{\mathfrak{B}}_{t} }}$ and $\sigma_{{{\mathfrak{B}}_{t} }}$ are functions that assign degrees of membership and non-membership, respectively. Moreover, the functions $\mu_{{{\mathfrak{B}}_{t} }}$ and $\sigma_{{{\mathfrak{B}}_{t} }}$ satisfy the condition: $0 \le \mu_{{{\mathfrak{B}}_{t} }} \left( {u_{1} } \right) + \sigma_{{{\mathfrak{B}}_{t} }} \left( {u_{1} } \right) \le 1.$

Definition 3 48

Let ${{\mathbb{G}}^{\prime}} = \langle{\mathbb{V}},{\mathbb{E}}\rangle$ be a simple graph. A pair ${\mathcal{G}} = \langle{\mathcal{A}},{\mathcal{B}}\rangle$ is said to be an intuitionistic fuzzy graph ( ${\mathbb{IFG}}$ ) on graph ${{\mathbb{G}}^{\prime}},$ where ${\mathcal{A}} = \left\{ { \left\langle { u_{i} , \mu_{{\mathcal{A}}} \left( {u_{i} } \right), \sigma_{{\mathcal{A}}} \left( {u_{i} } \right)} \right\rangle :u_{i} \in {\mathbb{V}}} \right\}$ is an ${\mathbb{IFS}}$ on ${\mathbb{V}}$ and $\mathcal{B}=\{ < {{u}}_{i}, {\mu }_{\mathcal{B}}\left({{u}}_{i},{{u}}_{j}\right), {\sigma }_{\mathcal{B}}\left({{u}}_{i},{{u}}_{j}\right)>:\left({{u}}_{i},{{u}}_{j}\right)\in {\mathbb{E}}\}$ is an ${\mathbb{IFS}}$ on ${\mathbb{E}} \subseteq {\mathbb{V}} \times {\mathbb{V}}$ such that for every edge $\left( {u_{i} ,u_{j} } \right) \in {\mathbb{E}}$ .

Satisfy the conditions: $0\le {\mu }_{\mathcal{A}}\left({{u}}_{i}\right)+{\sigma }_{\mathcal{A}}\left({{u}}_{i}\right)\le 1$ and $0\le {\mu }_{\mathcal{B}}\left({{u}}_{i},{{u}}_{j}\right)+{\sigma }_{\mathcal{B}}\left({{u}}_{i},{{u}}_{j}\right)\le 1.$

Definition 4 51

The order of ${\mathbb{IFG}}$ $\mathcal{G}$ is specified by:

Definition 5 51

The degree of a vertex ${{u}}_{1}$ in ${\mathbb{IFG}}$ $\mathcal{G}$ is given by:

t-intuitionistic fuzzy graph

This section defines a t-intuitionistic fuzzy graph and explores various fundamental properties of this phenomenon.

Definition 6

Let ${\mathcal{G}} = \left\langle {{\mathcal{A}},{ \mathcal{B}}} \right\rangle$ be an intuitionistic fuzzy graph ( ${\mathbb{IFG}}$ ) on a simple graph ${{\mathbb{G}}^{\prime}} = \left\langle {{\mathbb{V}},{\mathbb{E}}} \right\rangle$ . An ${\mathbb{IFG}}$ ${\mathcal{G}}$ is called a t-intuitionistic fuzzy graph (t- ${\mathbb{IFG}}$ ) is denoted by ${\mathcal{G}}_{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle ,$ where ${\mathcal{A}}_{t} = \left\{ {\left( {u_{i} , \mu_{{{\mathcal{A}}_{t} }} \left( {u_{i} } \right),\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{i} } \right)} \right):u_{i} \in {\mathbb{V}}} \right\}$ is a t- ${\mathbb{IFS}}$ on ${\mathbb{V}}$ and ${\mathcal{B}}_{t} = \left\{ {\left( {\left( {u_{i} ,u_{j} } \right), \mu_{{{\mathcal{B}}_{t} }} \left( {u_{i} ,u_{j} } \right),\sigma_{{{\mathcal{B}}_{t} }} \left( {u_{i} ,u_{j} } \right)} \right):\left( {u_{i} ,u_{j} } \right) \in {\mathbb{E}}} \right\}$ is a t- ${\mathbb{IFS}}$ on ${\mathbb{E}} \subseteq {\mathbb{V}} \times {\mathbb{V}},$ such that for every edge $\left( {u_{i} ,u_{j} } \right) \in {\mathbb{E}}$ .

Satisfy the conditions: $0\le {\mu }_{{\mathcal{A}}_{t}}\left({{u}}_{i}\right)+{\sigma }_{{\mathcal{A}}_{t}}\left({{u}}_{i}\right)\le 1$ and $0\le {\mu }_{{\mathcal{B}}_{t}}\left({{u}}_{i},{{u}}_{j}\right)+{\sigma }_{{\mathcal{B}}_{t}}\left({{u}}_{i},{{u}}_{j}\right)\le 1.$

Here ${\mu }_{{\mathcal{A}}_{t}}\left({{u}}_{i}\right)$ and ${\sigma }_{{\mathcal{A}}_{t}}\left({{u}}_{i}\right)$ represents the membership and non-membership degrees of nodes ${{u}}_{i}\in {\mathbb{V}}.$ The terms ${\mu }_{{\mathcal{B}}_{t}}\left({{u}}_{i},{{u}}_{j}\right)$ and ${\sigma }_{{\mathcal{B}}_{t}}\left({{u}}_{i},{{u}}_{j}\right)$ represents the membership and non-membership degrees of edges $({{u}}_{i},{{u}}_{j})\in {\mathbb{E}},$ respectively.

Consider a graph ${\mathbb{G}}^{\prime}=\langle {\mathbb{V}},{\mathbb{E}}\rangle$ such that.

The ${\mathbb{IFS}}$ ${\mathcal{A}}$ of ${\mathbb{V}}$ is given by:

The ${\mathbb{IFS}}$ ${\mathcal{B}}$ of ${\mathbb{E}}$ is given by:

The application of the Definition ( 2 ) on the two ${\mathbb{IFS}}$ ${\mathcal{A}}$ and ${\mathcal{B}}$ corresponding to the value $t = 0.70$ gives that:

The graphical representation of $0.70$ - ${\mathbb{IFG}}$ ${\mathcal{G}}_{0.7} = \left\langle {{\mathcal{A}}_{0.7} ,{\mathcal{B}}_{0.7} } \right\rangle$ is displayed in Fig. 1 .

$0.70 - {\mathbb{IFG}} \,{\mathcal{G}}_{0.7}$ .

Definition 7

A t- ${\mathbb{IFG}}$ ${\mathcal{H}}_{t} = \left\langle {{\mathcal{A}}_{t}^{\prime} ,{\mathcal{B}}_{t}^{\prime} } \right\rangle$ is said to be a t-intuitionistic fuzzy subgraph of t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ if ${\mathcal{A}}_{t}^{\prime} \subseteq {\mathcal{A}}_{t}$ and ${\mathcal{B}}_{t}^{\prime} \subseteq {\mathcal{B}}_{t} .$

Definition 8

A t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ is said to be complete t- ${\mathbb{IFG}}$ if it admits the following conditions:

Consider the complete $0.80$ - ${\mathbb{IFG}}$ ${\mathcal{G}}_{t}$ as depicted in Fig. 2 .

Complete $0.80 - {\mathbb{IFG}} \,{\mathcal{G}}_{0.80}$ .

Definition 9

Let ${\mathcal{G}}_{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ be a t- ${\mathbb{IFG}}$ $,$ where ${\mathcal{A}}_{t} = \left\langle {\mu_{{{\mathcal{A}}_{t} }} ,\sigma_{{{\mathcal{A}}_{t} }} } \right\rangle$ is a t- ${\mathbb{IFS}}$ on ${\mathbb{V}}$ and ${\mathcal{B}}_{t} = \left\langle {\mu_{{{\mathcal{B}}_{t} }} ,\sigma_{{{\mathcal{B}}_{t} }} } \right\rangle$ is a t- ${\mathbb{IFS}}$ on ${\mathbb{E}} \subseteq {\mathbb{V}} \times {\mathbb{V}}.$ Then.

The order of t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{t}$ is defined as:

The size of t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{t}$ is defined as:

The order of t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{t}$ is $\left( {3.9,2} \right).$ (see Example 1 ).

Proposition 1

Let ${\mathcal{G}}_{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ be a t- ${\mathbb{IFG}}$ $,$ then ${\mathcal{S}}\left( {{\mathcal{G}}_{t} } \right) \le {\mathcal{O}}\left( {{\mathcal{G}}_{t} } \right).$

Definition 10

Let ${\mathcal{G}}_{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ be a t- ${\mathbb{IFG}}$ on ${\mathbb{G}} = \left\langle {{\mathbb{V}},{\mathbb{E}}} \right\rangle ,$ then:

In t- ${\mathbb{IFG}}\,\,{\mathcal{G}}_{t}$ , the degree of a vertex $u_{1}$ in ${\mathcal{G}}_{t}$ is defined as follows:

The minimum degree $\delta \left( {{\mathcal{G}}_{t} } \right)$ of t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{t}$ is given by:

The maximum degree $\Delta \left( {{\mathcal{G}}_{t} } \right)$ of ${\mathcal{G}}_{t}$ is defined as follows:

Proposition 2.

In t- ${\mathbb{IFG}}$ ${ \mathcal{G}}_{t} ,$ then the following inequality holds:

From Example 1 , the degree of each vertex in ${\mathcal{G}}_{t}$ are:

$deg_{{{\mathcal{G}}_{t} }} \left( a \right) = \left( {1.4,1} \right)$ , $deg_{{{\mathcal{G}}_{t} }} \left( b \right) = \left( {0.7,0.6} \right)$ , $deg_{{{\mathcal{G}}_{t} }} \left( c \right) = \left( {1,8.1.4} \right)$ , $deg_{{{\mathcal{G}}_{t} }} \left( d \right) = \left( {1,0.7} \right)$ , $deg_{{{\mathcal{G}}_{t} }} \left( e \right) = \left( {1.5,1.1} \right)$ , and $deg_{{{\mathcal{G}}_{t} }} \left( f \right) = \left( {1.2,0.6} \right)$ .

Let ${\mathcal{G}}_{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ be any t- ${\mathbb{IFG}}$ , then:

Let ${\mathcal{G}}_{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ be a t- ${\mathbb{IFG}}$ .

The application of the part (1) of Definition ( 9 ) to gives that:

Hence, it completes the proof.

Corollary 1.

In a t- ${\mathbb{IFG}}$ , the odd membership degree and the odd non-membership degree have an even number of vertices.

Corollary 2.

In a t- ${\mathbb{IFG}}$ , $n - 1$ is the maximum degree of any vertex $n$ .

Operations on t-intuitionistic fuzzy graph

This section explores the set-theoretical operations of t- ${\mathbb{IFG}}$ $.$ We also establish and analyze the fundamental characteristics of these phenomena.

Definition 11.

Let ${\mathcal{G}}_{1}^{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ and ${\mathcal{G}}_{2}^{t} = \left\langle {{\mathcal{A}}_{t}^{\prime} ,{\mathcal{B}}_{t}^{\prime} } \right\rangle$ be any two t- ${\mathbb{IFG}}$ of ${\mathbb{G}}_{1} = \left\langle {{\mathbb{V}},{\mathbb{E}}} \right\rangle$ and ${\mathbb{G}}_{2} = \left\langle {{\mathbb{V}}^{\prime}},{{\mathbb{E}}^{\prime}} \right\rangle ,$ respectively. The Cartesian product ${\mathcal{G}}_{1}^{t} \times {\mathcal{G}}_{2}^{t}$ of t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ is defined by $\left\langle {{\mathcal{A}}_{t} \times {\mathcal{A}}_{t}^{\prime} ,{\mathcal{B}}_{t} \times {\mathcal{B}}_{t}^{\prime} } \right\rangle$ , where ${\mathcal{A}}_{t} \times {\mathcal{A}}_{t}^{\prime}$ and ${\mathcal{B}}_{t} \times {\mathcal{B}}_{t}^{\prime}$ are t- ${\mathbb{IFS}}$ on ${\mathbb{V}} \times {{\mathbb{V}}^{\prime}} = \{ \left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right):u_{1} ,u_{2} \in {\mathbb{V}} \wedge w_{1} ,w_{2} \in {{\mathbb{V}}^{\prime}}$ } and ${\mathbb{E}} \times {{\mathbb{E}}^{\prime}} = \left\{ {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right):u_{1} = u_{2} ,u_{1} ,u_{2} \in {\mathbb{V}},\left( {w_{1} ,w_{2} } \right) \in {{\mathbb{E}}^{\prime}}} \right\} \cup \left\{ {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right):w_{1} = w_{2} ,w_{1} ,w_{2} \in {{\mathbb{V}}^{\prime}} ,\left( {u_{1} ,u_{2} } \right) \in {\mathbb{E}}} \right\} \cup \left\{ {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right):w_{1} \ne w_{2} ,w_{1} ,w_{2} \in {{\mathbb{V}}^{\prime}} ,\left( {u_{1} ,u_{2} } \right) \in {\mathbb{E}}} \right\},$ respectively, which satisfies the following conditions:

$\forall \left( {u_{1} ,w_{1} } \right) \in {\mathbb{V}} \times {{\mathbb{V}}^{\prime}}$

$\mu_{{{\mathcal{A}}_{t} \times {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = min\left\{ {\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

$\sigma_{{{\mathcal{A}}_{t} \times {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = max\left\{ {\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

If $u_{1} = u_{2}$ and $\forall \left( {w_{1} ,w_{2} } \right) \in {{\mathbb{E}}^{\prime}}$

$\mu_{{{\mathcal{B}}_{t} \times {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right)} \right) = min\left\{ {\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\mu_{{{\mathcal{B}}_{t}^{\prime} }} \left( {w_{1} ,w_{2} } \right)} \right\}$

$\sigma_{{{\mathcal{B}}_{t} \times {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right)} \right) = max\left\{ {\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathcal{B}}_{t}^{\prime} }} \left( {w_{1} ,w_{2} } \right)} \right\}$

If $w_{1} = w_{2}$ and $\forall \left( {u_{1} ,u_{2} } \right) \in {\mathbb{E}}$

$\mu_{{{\mathcal{B}}_{t} \times {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right)} \right) = min\left\{ {\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,u_{2} } \right),\mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

$\sigma_{{{\mathcal{B}}_{t} \times {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right)} \right) = max\left\{ {\sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,u_{2} } \right),\sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

Consider the two $0.8$ - ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ illustrated in Figs. 3 and 4 .

$0.8 - {\mathbb{IFG}} \,{\mathcal{G}}_{1}^{0.8}$ .

$0.8 - {\mathbb{IFG}} \,{\mathcal{G}}_{2}^{0.8}$ .

Figure 5 shows their corresponding Cartesian product ${\mathcal{G}}_{1}^{0.8} \times {\mathcal{G}}_{2}^{0.8} :$

$0.8 -$ ${\mathbb{IFG}}$ of ${\mathcal{G}}_{1}^{0.8} \times {\mathcal{G}}_{2}^{0.8}$ .

Definition 12.

The degree of a vertex in ${\mathcal{G}}_{1}^{t} \times {\mathcal{G}}_{2}^{t}$ is defined as follows: for any $\left( {u_{1} ,w_{1} } \right) \in {\mathbb{V}} \times {{\mathbb{V}}^{\prime}}$ .

According to Example 5 , each vertex in ${\mathcal{G}}_{1}^{t} \times {\mathcal{G}}_{2}^{t} \user2{ }$ has the following degree:

Proposition 3.

The Cartesian product of two t- ${\mathbb{IFG}}$ is a t- ${\mathbb{IFG}}$ $.$

The condition for ${\mathcal{A}}_{t} \times {\mathcal{A}}_{t}^{\prime}$ is self-explanatory. Let $u_{1} \in {\mathbb{V}}$ and $\left( {w_{1} ,w_{2} } \right) \in {{\mathbb{E}}^{\prime}}.$ Then:

Consequently $\mu_{{{\mathcal{B}}_{t} \circ {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{1} ,w_{2} } \right)} \right) \le min\left\{ {\mu_{{{\mathcal{A}}_{t} \times {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right),\mu_{{{\mathcal{A}}_{t} \times {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} ,w_{2} } \right)} \right\}, {\text{if}} \;u_{1} \in {\mathbb{V}}$ and $\left( {w_{1} ,w_{2} } \right) \in {{\mathbb{E}}^{\prime}}.$

Thus $\sigma_{{{\mathcal{B}}_{t} \circ {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{1} ,w_{2} } \right)} \right) \le max\left\{ {\sigma_{{{\mathcal{A}}_{t} \times {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right),\sigma_{{{\mathcal{A}}_{t} \times {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} ,w_{2} } \right)} \right\},{\text{if}}\;u{\ominus }_{1} \in {\mathbb{V}}$ and $\left( {w_{1} ,w_{2} } \right) \in {{\mathbb{E}}^{\prime}}.$

Likewise, we can demonstrate it for $w_{1} \in {{\mathbb{V}}^{\prime}}$ and $\left( {u_{1} ,u_{2} } \right) \in {\mathbb{E}}$ .

Definition 13.

The composition ${\mathcal{G}}_{1}^{t} \circ {\mathcal{G}}_{2}^{t}$ of two t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ is a t- ${\mathbb{IFG}}$ and defined as a pair $\left\langle {{\mathcal{A}}_{t} \circ {\mathcal{A}}_{t}^{\prime} ,{\mathcal{B}}_{t} \circ {\mathcal{B}}_{t}^{\prime} } \right\rangle$ , where ${\mathcal{A}}_{t} \circ {\mathcal{A}}_{t}^{\prime}$ and ${\mathcal{B}}_{t} \circ {\mathcal{B}}_{t}^{\prime}$ are t- ${\mathbb{IFS}}$ on ${\mathbb{V}} \times {{\mathbb{V}}^{\prime}} = \{ \left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right):u_{1} ,u_{2} \in {\mathbb{V}} \wedge w_{1} ,w_{2} \in {{\mathbb{V}}^{\prime}}$ } and ${\mathbb{E}} \times {{\mathbb{E}}^{\prime}} = \left\{ {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right):u_{1} = u_{2} ,u_{1} ,u_{2} \in {\mathbb{V}},\left( {w_{1} ,w_{2} } \right) \in {{\mathbb{E}}^{\prime}}} \right\} \cup \left\{ {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right):w_{1} = w_{2} ,w_{1} ,w_{2} \in {{\mathbb{V}}^{\prime}} ,\left( {u_{1} ,u_{2} } \right) \in {\mathbb{E}}} \right\} \cup \left\{ {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right):w_{1} \ne w_{2} ,w_{1} ,w_{2} \in {{\mathbb{V}}^{\prime}} ,\left( {u_{1} ,u_{2} } \right) \in {\mathbb{E}}} \right\},$ respectively, which satisfies the following conditions:

$\mu_{{{\mathcal{A}}_{t} \circ {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = min\left\{ {\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

$\sigma_{{{\mathcal{A}}_{t} \circ {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = max\left\{ {\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

If $u_{1} = u_{2}$ and $\forall (w_{1} ,w_{2} ) \in {{\mathbb{E}}^{\prime}}$

$\mu_{{{\mathcal{B}}_{t} \circ {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right)} \right) = min\left\{ {\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\mu_{{{\mathcal{B}}_{t}^{\prime} }} \left( {w_{1} ,w_{2} } \right)} \right\}$

$\sigma_{{{\mathcal{B}}_{t} \circ {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right)} \right) = max\left\{ {\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathcal{B}}_{t}^{\prime} }} \left( {w_{1} ,w_{2} } \right)} \right\}$

If $w_{1} = w_{2}$ and $\forall (u_{1} ,u_{2} ) \in {\mathbb{E}}$

$\mu_{{{\mathcal{B}}_{t} \circ {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right)} \right) = min\left\{ {\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,u_{2} } \right),\mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

$\sigma_{{{\mathcal{B}}_{t} \circ {\mathcal{B}}_{t}^{\prime} }} \left( {\left( {u_{1} ,w_{1} ),(u_{2} ,w_{2} } \right)} \right) = max\left\{ {\sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,u_{2} } \right),\sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

If $w_{1} \ne w_{2}$ and $\forall (u_{1} ,u_{2} ) \in {\mathbb{E}}$

Consider the two $0.9$ - ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ as shown in Figs. 6 and 7 .

$0.9 - {\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{0.9}$ .

$0.9 - {\mathbb{IFG}}$ ${\mathcal{G}}_{2}^{0.9}$ .

Then, their corresponding composition ${\mathcal{G}}_{1}^{t} \circ {\mathcal{G}}_{2}^{t}$ is shown in Fig. 8 .

Graphical representation of ${\mathcal{G}}_{1}^{0.9} \circ {\mathcal{G}}_{2}^{0.9}$ .

Definition 14.

The degree of a vertex in ${\mathcal{G}}_{1}^{t} \circ {\mathcal{G}}_{2}^{t}$ is defined as follows: for any $\left( {u_{1} ,w_{1} } \right) \in {\mathbb{V}} \times {{\mathbb{V}}^{\prime}}$ .

From Example 7 , the degree of each vertex in ${\mathcal{G}}_{1}^{t} \circ {\mathcal{G}}_{2}^{t}$ are:

$deg_{{{\mathcal{G}}_{1}^{t} \circ {\mathcal{G}}_{2}^{t} }} \left( {a,u} \right) = \left( {0.9,1.4} \right)$ , $deg_{{{\mathcal{G}}_{1}^{t} \circ {\mathcal{G}}_{2}^{t} }} \left( {a,v} \right) = \left( {0.9,1.4} \right)$ , $deg_{{{\mathcal{G}}_{1}^{t} \circ {\mathcal{G}}_{2}^{t} }} \left( {b,u} \right) = \left( {0.9,1.6} \right)$ , and $deg_{{{\mathcal{G}}_{1}^{t} \circ {\mathcal{G}}_{2}^{t} }} \left( {b,v} \right) = \left( {0.9,1.6} \right)$ .

Proposition 4.

Let ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ be any two t- ${\mathbb{IFG}}$ then ${\mathcal{G}}_{1}^{t} \circ {\mathcal{G}}_{2}^{t}$ is also a t- ${\mathbb{IFG}}$ $.$

Definition 15.

If $u_{1} \in {\mathbb{V}}$ and $u_{1} \notin {{\mathbb{V}}^{\prime}}$

$\mu_{{{\mathcal{A}}_{t} \cup {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = \mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right)$

$\sigma_{{{\mathcal{A}}_{t} \cup {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = \sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right)$

If $u_{1} \notin {\mathbb{V}}$ and $u_{1} \in {{\mathbb{V}}^{\prime}}$

$\mu_{{{\mathcal{A}}_{t} \cup {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = \mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right)$

$\sigma_{{{\mathcal{A}}_{t} \cup {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = \sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right)$

If $u_{1} \in {\mathbb{V}} \cap {{\mathbb{V}}^{\prime}}$

$\mu_{{{\mathcal{A}}_{t} \cup {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = max\left\{ {\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right)} \right\}$

$\sigma_{{{\mathcal{A}}_{t} \cup {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = min\left\{ {\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right)} \right\}$

If $\left( {u_{1} ,w_{1} } \right) \in {\mathbb{E}}$ and $\left( {u_{1} ,w_{1} } \right) \notin {{\mathbb{E}}^{\prime}}$

$\mu_{{{\mathcal{B}}_{t} \cup {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = \mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right)$

$\sigma_{{{\mathcal{B}}_{t} \cup {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = \sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right)$

If $\left( {u_{1} ,w_{1} } \right) \notin {\mathbb{E}}$ and $\left( {u_{1} ,w_{1} } \right) \in {{\mathbb{E}}^{\prime}}$

$\mu_{{{\mathcal{B}}_{t} \cup {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = \mu_{{{\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right)$

$\sigma_{{{\mathcal{B}}_{t} \cup {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = \sigma_{{{\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right)$

If $\left( {u_{1} ,w_{1} } \right) \in {\mathbb{E}} \cap {{\mathbb{E}}^{\prime}}$

$\mu_{{{\mathcal{B}}_{t} \cup {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = max\left\{ {\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right),\mu_{{{\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right)} \right\}$

$\sigma_{{{\mathcal{B}}_{t} \cup {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = min\left\{ {\sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right),\sigma_{{{\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right)} \right\}.$

Consider the two $0.9 - {\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ as shown in Figs. 9 and 10 .

$0.9 - {\mathbb{IFG}} \,{\mathcal{G}}_{1}^{0.9}$ .

$0.9 - {\mathbb{IFG}} \,{\mathcal{G}}_{2}^{0.9}$ .

Figure 11 depicts the graphical representation of the union ${\mathcal{G}}_{1}^{0.9} \cup {\mathcal{G}}_{2}^{0.9}$ of two 0.9- ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{0.9}$ and ${\mathcal{G}}_{2}^{0.9} .$

Graphical representation of ${\mathcal{G}}_{1}^{0.9} \cup {\mathcal{G}}_{2}^{0.9}$ .

Definition 16.

The following formula describes the degree of a vertex $\left( {u_{1} ,w_{1} } \right)$ at a t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t} \cup {\mathcal{G}}_{2}^{t} :$ For any $\left( {u_{1} ,w_{1} } \right) \in {\mathbb{V}} \times {{\mathbb{V}}^{\prime}}$ .

Proposition 5.

The union of two t- ${\mathbb{IFG}}$ is also a t- ${\mathbb{IFG}}$ $.$

Definition 17.

$\mu_{{{\mathcal{A}}_{t} + {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = \mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right)$

$\sigma_{{{\mathcal{A}}_{t} + {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = \sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right)$

$\mu_{{{\mathcal{A}}_{t} + {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = \mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right)$

$\sigma_{{{\mathcal{A}}_{t} + {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = \sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right)$

$\mu_{{{\mathcal{A}}_{t} + {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = max\left\{ {\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right)} \right\}$

$\sigma_{{{\mathcal{A}}_{t} + {\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right) = min\left\{ {\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {u_{1} } \right)} \right\}$

$\mu_{{{\mathcal{B}}_{t} + {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = \mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right)$

$\sigma_{{{\mathcal{B}}_{t} + {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = \sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right)$

$\mu_{{{\mathcal{B}}_{t} + {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = \mu_{{{\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right)$

$\sigma_{{{\mathcal{B}}_{t} + {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = \sigma_{{{\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right)$

If $\left( v \right) \in {\mathbb{E}} \cap {{\mathbb{E}}^{\prime}}$

$\mu_{{{\mathcal{B}}_{t} + {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = max\left\{ {\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right),\mu_{{{\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right)} \right\}$

$\sigma_{{{\mathcal{B}}_{t} + {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = min\left\{ {\sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right),\sigma_{{{\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right)} \right\}.$

If $\left( {u_{1} ,w_{1} } \right) \in {{\mathbb{E}}^{\prime\prime}}$

$\mu_{{{\mathcal{B}}_{t} + {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = min\left\{ {\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

$\sigma_{{{\mathcal{B}}_{t} + {\mathcal{B}}_{t}^{\prime} }} \left( {u_{1} ,w_{1} } \right) = max\left\{ {\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {w_{1} } \right)} \right\}$

Example 10.

From Example 9 , the graphical representation of $0.9 - {\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t} + {\mathcal{G}}_{2}^{t}$ of ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ as shown in Fig. 12 .

Graphical representation of ${\mathcal{G}}_{1}^{0.9} + {\mathcal{G}}_{2}^{0.9}$ .

Definition 18.

Let ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ be any two $t - {\mathbb{IFG}}$ . The degree of a vertex in t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t} + {\mathcal{G}}_{2}^{t}$ is defined as follows: for any $\left( {u_{1} ,w_{1} } \right) \in {\mathbb{V}} \times {{\mathbb{V}}^{\prime}}$ .

Proposition 6.

The join of two t- ${\mathbb{IFG}}$ is also a t- ${\mathbb{IFG}}$ $.$

Suppose that ${\mathcal{G}}_{1}^{t} \cup {\mathcal{G}}_{2}^{t}$ is a t- ${\mathbb{IFG}}$ . Let $\left( {u_{1} ,w_{1} } \right) \in {\mathbb{E}},$ $\left( {u_{1} ,w_{1} } \right) \notin {{\mathbb{E}}^{\prime}}$ and $u_{1} ,w_{1} \in {\mathbb{V}} - {{\mathbb{V}}^{\prime}}$ .

Consequently $\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right) \le min\left\{ {\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\mu_{{{\mathcal{A}}_{t} }} \left( {w_{1} } \right)} \right\}.$

Consequently $\sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right) \le max\left\{ {\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathcal{A}}_{t} }} \left( {w_{1} } \right)} \right\}.$

This shows that ${\mathcal{G}}_{1}^{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ is a t- ${\mathbb{IFG}}$ . In the same way, we obtain that ${\mathcal{G}}_{2}^{t} = \left\langle {{\mathcal{A}}_{t}^{\prime} ,{\mathcal{B}}_{t}^{\prime} } \right\rangle$ is a t- ${\mathbb{IFG}}$ of ${{\mathbb{G}}^{\prime\prime}}.$ Conversely, suppose that ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ are t- ${\mathbb{IFG}}$ $.$ We know that the union of two t- ${\mathbb{IFG}}$ is a t- ${\mathbb{IFG}}$ . Thus, ${\mathcal{G}}_{1}^{t} \cup {\mathcal{G}}_{2}^{t}$ is a t- ${\mathbb{IFG}}$ $.$

The proof for this is similar to the proof presented in Theorem 2 .

Isomorphism of t-intuitionistic fuzzy graphs

This section introduces the concepts of homomorphism and isomorphism of t- ${\mathbb{IFG}}$ and explores the essential properties of these ideas.

Definition 19.

$\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right) \le \mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {\theta \left( {u_{1} } \right)} \right)$ and $\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right) \le \sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {\theta \left( {u_{1} } \right)} \right)$ , $\forall u_{1} \in {\mathbb{V}}$

$\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right) \le \mu_{{{\mathcal{B}}_{t}^{\prime} }} \left( {\theta \left( {u_{1} } \right),\theta \left( {w_{1} } \right)} \right)$ and $\sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right) \le \sigma_{{{\mathcal{B}}_{t}^{\prime} }} \left( {\theta \left( {u_{1} } \right),\theta \left( {w_{1} } \right)} \right)$ , $\forall (u_{1} ,w_{1} ) \in {\mathbb{E}}.$

Definition 20.

A weak isomorphism $\theta$ from t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ to ${\mathcal{G}}_{2}^{t}$ is a bijective mapping $\theta :{\mathbb{V}} \to {{\mathbb{V}}^{\prime}},$ which meets the following conditions:

Definition 21.

$\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right) = \mu_{{{\mathcal{B}}_{t}^{\prime} }} \left( {\theta \left( {u_{1} } \right),\theta \left( {w_{1} } \right)} \right)$ and $\sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,w_{1} } \right) = \sigma_{{{\mathcal{B}}_{t}^{\prime} }} \left( {\theta \left( {u_{1} } \right),\theta \left( {w_{1} } \right)} \right)$ , $\forall \left( {u_{1} ,w_{1} } \right) \in {\mathbb{E}}.$

Definition 22.

An isomorphism between t- ${\mathbb{IFG}}$ s ${\mathcal{G}}_{1}^{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ and ${\mathcal{G}}_{2}^{t} = \left\langle {{\mathcal{A}}_{t}^{\prime} ,{\mathcal{B}}_{t}^{\prime} } \right\rangle$ is a bijective homomorphism mapping $\theta :{\mathbb{V}} \to {{\mathbb{V}}^{\prime}}$ (written as ${\mathcal{G}}_{1}^{t} \approx {\mathcal{G}}_{2}^{t} )$ which satisfies the following conditions:

$\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right) = \mu_{{{\mathcal{A}}_{t}^{\prime} }} \left( {\theta \left( {u_{1} } \right)} \right)$ and $\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right) = \sigma_{{{\mathcal{A}}_{t}^{\prime} }} \left( {\theta \left( {u_{1} } \right)} \right)$ , $\forall u_{1} \in {\mathbb{V}}$

Example 11.

Consider the two $0.8 - {\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ as shown in Figs. 13 and 14 .

$0.8 - {\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{0.8}$ .

The mapping $\zeta :{\mathbb{V}} \to {{\mathbb{V}}^{\prime}}$ is defined by $\zeta \left( a \right) = g,\zeta \left( b \right) = f$ and $\zeta \left( c \right) = e.$ Given Definition ( 22 ), we have ${\mathcal{G}}_{1}^{0.8} \approx {\mathcal{G}}_{2}^{0.8} .$

An isomorphism between t- ${\mathbb{IFG}}$ is an equivalence relation.

Reflexivity and symmetry are obvious. Let $\varphi :{\mathbb{V}} \to {{\mathbb{V}}^{\prime}}$ and $\theta :{{\mathbb{V}}^{\prime}} \to {{\mathbb{V}}^{\prime\prime}}$ be the isomorphisms of ${\mathcal{G}}_{1}^{t}$ onto ${\mathcal{G}}_{2}^{t}$ and ${\mathcal{G}}_{2}^{t}$ onto ${\mathcal{G}}_{3}^{t}$ respectively.

Then $\theta \circ \varphi :{\mathbb{V}} \to {{\mathbb{V}}^{\prime\prime}}$ is a bijective map from ${\mathbb{V}}$ to ${{\mathbb{V}}^{\prime\prime}}$ is defined as:

Since a map $\varphi :{\mathbb{V}} \to {{\mathbb{V}}^{\prime}}$ defined by $\varphi \left( {u_{1} } \right) = w_{1} ,\forall u_{1} \in {\mathbb{V}}$ is an isomorphism.

In view of Definition ( 22 ), we have

In the same way, we obtained that

By using the relations ( 1 ), ( 5 ) and $\varphi \left( {u_{1} } \right) = w_{1} ,u_{1} \in {\mathbb{V}},$ we have

Similarly, we obtain that $\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right) = \sigma_{{{\mathcal{A}}_{t}^{^{\prime\prime}} }} \left( {\theta \left( {\varphi \left( {u_{1} } \right)} \right)} \right).$

When relations ( 3 ) and ( 7 ) are applied, the outcome is that:

Similarly, we find that $\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,u_{2} } \right) = \mu_{{{\mathcal{B}}_{t}^{^{\prime\prime}} }} \left( {\theta (\varphi (u_{1} )),\theta (\varphi (u_{2} ))} \right).$

Thus, $\theta \circ \varphi$ is an isomorphism between ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{3}^{t} .$

Complement of t-intuitionistic fuzzy graph

This section defines the concept of a complement of t- ${\mathbb{IFG}}$ and investigates its essential features.

Definition 23.

Let ${\mathcal{G}}_{1}^{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ be a t- ${\mathbb{IFG}}$ of ${{\mathbb{G}}^{\prime}} = \left\langle {{\mathbb{V}},{\mathbb{E}}} \right\rangle$ . The complement of a t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ is a t- ${\mathbb{IFG}}$ $\overline{{{\mathcal{G}}_{1}^{t} }}$ on $\overline{{{{\mathbb{G}}^{\prime}}}} = \left\langle {{\overline{\mathbb{V}}},{\overline{\mathbb{E}}}} \right\rangle$ is defined as follows:

${\overline{\mathbb{V}}} = {\mathbb{V}}$

If $u_{1} \in {\mathbb{V}}$ then $\mu_{{\overline{{{\mathcal{A}}_{t} }} }} \left( {u_{1} } \right) = \mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right)$ and $\sigma_{{\overline{{{\mathcal{A}}_{t} }} }} \left( {u_{1} } \right) = \sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right)$

If $\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,u_{2} } \right) \ne 0$ and $\sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,u_{2} } \right) \ne 0$ then $\mu_{{\overline{{{\mathcal{B}}_{t} }} }} \left( {u_{1} ,u_{2} } \right) = 0 = \sigma_{{\overline{{{\mathcal{B}}_{t} }} }} \left( {u_{1} ,u_{2} } \right)$

If $\mu_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,u_{2} } \right) = 0 = \sigma_{{{\mathcal{B}}_{t} }} \left( {u_{1} ,u_{2} } \right) = 0$ then $\mu_{{\overline{{{\mathcal{B}}_{t} }} }} \left( {u_{1} ,u_{2} } \right) = min\left\{ {\mu_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\mu_{{{\mathcal{A}}_{t} }} \left( {u_{2} } \right)} \right\}$ and $\sigma_{{\overline{{{\mathcal{B}}_{t} }} }} \left( {u_{1} ,u_{2} } \right) = max\left\{ {\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{1} } \right),\sigma_{{{\mathcal{A}}_{t} }} \left( {u_{2} } \right)} \right\}$

Example 12.

Consider a $0.8 - {\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ as shown in Fig. 15 .

$0.8 -$ ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{0.8}$ .

Then the complement $\overline{{{\mathcal{G}}_{1}^{t} }}$ of $0.8 - {\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ is shown in Fig. 16 .

$0.8 -$ ${\mathbb{IFG}}$ $\overline{{{\mathcal{G}}_{1}^{0.8} }}$ .

Definition 24.

A t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ is called self-complementary t- ${\mathbb{IFG}}$ if $\overline{{{\mathcal{G}}_{1}^{t} }} \approx {\mathcal{G}}_{1}^{t} .$

Proposition 7.

Let ${\mathcal{G}}_{1}^{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ be a self-complementary t- ${\mathbb{IFG}}$ $.$ Then

Proposition 8.

Let ${\mathcal{G}}_{1}^{t} = \left\langle {{\mathcal{A}}_{t} ,{\mathcal{B}}_{t} } \right\rangle$ be a t- ${\mathbb{IFG}}$ . If

Then ${\mathcal{G}}_{1}^{t}$ is a self-complementary t- ${\mathbb{IFG}}$ $.$

Proposition 9.

If ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ are two t- ${\mathbb{IFG}}$ such that ${\mathbb{V}} \cap {{\mathbb{V}}^{\prime}} = \emptyset$ then $\overline{{{\mathcal{G}}_{1}^{t} + {\mathcal{G}}_{2}^{t} }} \cong \overline{{{\mathcal{G}}_{1}^{t} }} \cup \overline{{{\mathcal{G}}_{2}^{t} }} .$

Proposition 10.

If ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ are two t- ${\mathbb{IFG}}$ such that ${\mathbb{V}} \cap {{\mathbb{V}}^{\prime}} = \emptyset$ then $\overline{{{\mathcal{G}}_{1}^{t} \cup {\mathcal{G}}_{2}^{t} }} \cong \overline{{{\mathcal{G}}_{1}^{t} }} + \overline{{{\mathcal{G}}_{2}^{t} }} .$

Proposition 11.

For any two t- ${\mathbb{IFG}}$ ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t} .$ If ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ have a strong isomorphism, then $\overline{{{\mathcal{G}}_{1}^{t} }}$ and $\overline{{{\mathcal{G}}_{2}^{t} }}$ also have a strong isomorphism.

Let $\varphi$ be a strong isomorphism between ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ . Since $\varphi$ is a bijective map, then $\varphi^{ - 1}$ is also a bijective map such that $\varphi^{ - 1} \left( {w_{1} } \right) = u_{1}{_{1}} ,\forall q_{1} \in {{\mathbb{V}}^{\prime}}$ . Thus.

By employing Definition ( 23 ), it becomes evident that:

Thus $\mu_{{\overline{{{\mathcal{B}}_{t} }} }} \left( {u_{1} ,w_{1} } \right) \le \mu_{{\overline{{{\mathcal{B}}_{t}^{\prime} }} }} \left( {u_{2} ,w_{2} } \right)$ .

Consequently $\sigma_{{\overline{{{\mathcal{B}}_{t} }} }} \left( {u_{1} ,w_{1} } \right) \le \sigma_{{\overline{{{\mathcal{B}}_{t}^{\prime} }} }} \left( {u_{2} ,w_{2} } \right)$ .

This shows that $\varphi^{ - 1}$ is a strong isomorphism between $\overline{{{\mathcal{G}}_{1}^{t} }}$ and $\overline{{{\mathcal{G}}_{2}^{t} }} .$

Proposition 12.

Let ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ be two t- ${\mathbb{IFG}}$ $.$ Then ${\mathcal{G}}_{1}^{t} \approx {\mathcal{G}}_{2}^{t}$ if and only if $\overline{{{\mathcal{G}}_{1}^{t} }} \approx \overline{{{\mathcal{G}}_{2}^{t} }} .$

Proposition 13.

Let ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t}$ be two t- ${\mathbb{IFG}}$ $.$ If there is a co-strong isomorphism between ${\mathcal{G}}_{1}^{t}$ and ${\mathcal{G}}_{2}^{t} ,$ then there is a homomorphism between $\overline{{{\mathcal{G}}_{1}^{t} }}$ and $\overline{{{\mathcal{G}}_{2}^{t} }} .$

Application of t-intuitionistic fuzzy graph

This section applies the theory of t- ${\mathbb{IFG}}$ to the decision-making process of alleviating poverty .

Developing nations have been profoundly affected by extreme destitution, which has significantly impacted their economies, societies, and a vast number of people globally. The escalation in the poverty rate can be attributed to various factors. Poverty is characterized by the inability to provide oneself and one's family with necessities such as food, clothing, and shelter. It can be examined from psychological, social, political, and economic perspectives. These circumstances can lead to criminal activity, drug abuse, and even fatalities. To address poverty reduction effectively, the t- ${\mathbb{IFG}}$ provides a mathematical representation and analysis of uncertain data. By utilizing the t- ${\mathbb{IFG}}$ , we can model and analyze elements related to poverty alleviation. This approach enables us to identify the most crucial variables in systematically and organized eliminating poverty, enhancing decision-making in poverty reduction efforts. Reducing poverty requires a multifaceted strategy that addresses the underlying causes of poverty and implements interventions designed to alleviate it. Some main factors are beneficial in reducing poverty, such as promoting economic growth $\left( {f_{1} } \right),$ creating employment opportunities $(f_{2} ),$ enhancing access to education and skills training $(f_{3} ),$ promoting manufacturing sectors $(f_{4} ),$ promoting industrialization $(f_{5} ),$ improving agriculture $(f_{6} ),$ and improving infrastructure $(f_{7} ).$ Let ${\mathbb{V}} = \left\{ {f_{1} ,f_{2} ,f_{3} ,f_{4} ,f_{5} ,f_{6} ,f_{7} } \right\}$ represents the vertex of the set of factors that significantly contribute to the fight against poverty. Let the edges depict the degree of connection or relationship between the factors as t-intuitionistic fuzzy values. The graphical representation of the factor of reduction poverty is displayed in Fig. 17 . Within the poverty reduction framework, membership and non-membership functions in intuitionistic fuzzy logic denote the connection between two distinct items or factors. These functions capture the degree to which an element exhibits membership or non-membership in a particular factor, facilitating a complete understanding of their interconnections. Within the poverty reduction framework, the membership function concept pertains to the extent to which an element exhibits favorable alignment with a particular factor. The non-membership function is a measure that measures the extent to which an element deviates from or lacks affiliation with a particular factor. The integration of these two functions offers a comprehensive understanding of the correlation between an element and a particular factor, encompassing its positive correlation and divergence from said factor. Decision-makers can evaluate the intricate and uncertain connections between different aspects of poverty reduction efforts by considering membership and non-membership functions. The parameter 't' allows decision-makers to customize the t- ${\mathbb{IFG}}$ according to their domain knowledge and problem.

Graphical representation of poverty reduction factors.

Moreover, different parameter values of 't' indicate different attitudes towards risk and uncertainty. The direction of threshold values for membership and non-membership allows decision-makers to highlight or de-emphasize certain facts depending on their desired degree of membership and non-membership. The parameter denoted as 't' enables the adaptation of t- ${\mathbb{IFG}}$ to different contexts and sensitivities. Adjusting variable 't' allows decision-makers to explore various possibilities by manipulating the balance between positive and opposing viewpoints. This ability is critical when making decisions in uncertain contexts or during ongoing changes.

The ${\mathbb{IFS}}$ and $0.8 -$ ${\mathbb{IFS}}$ defined on the edges is shown in the following Table 1 .

In Table 1 , the edge $e_{12}$ from “promoting economic growth” to "creating employment opportunities" indicates that promoting economic growth is related to creating more job opportunities. In edge $e_{12} = \left( {0.8,0.2} \right)$ , the membership degree 0.8 indicates a strong connection between these factors, and the non-membership degree 0.2 shows a weak connection between these factors in reducing poverty. In the same way, an edge $e_{36}$ from "improving access to education and skills training" to "improving agriculture" shows that better education and skills training can lead to better farming practices and higher productivity. In the given context, the membership degree of 0.7 for the edge $e_{36} = \left( {0.7,0.3} \right)$ signifies a significant correlation or positive influence between the factors to reduce poverty. Conversely, the non-membership degree of 0.3 suggests a relatively low perception of disassociation or lack of relevance between these factors and poverty reduction efforts. Here, the parameter “ t ” suggests which factor can reduce poverty by $80\%$ .

As shown in Table 2 , the application of part $\left( 1 \right)$ of Definition ( 10 ) yields the following results.

The score function of the edges is defined as:

The score function of the edges is calculated to find the optimal factor.

The results shown in Table 3 are then obtained by using the score function formula from Table 2 .

Figure 18 depicts the graphical representation of the score function for the factors listed in Table 3 .

Graphical representation of score function of factor.

Consequently, ${\mathfrak{T}}\left( {f_{1} } \right) = 4.3566$ is the greatest value, and according to the parameter $t$ , $f_{1}$ is the most significant factor in reducing poverty. Promoting economic growth can create jobs, raise incomes, encourage entrepreneurship and new ideas, lower the prices of goods and services, and give governments more money to spend on social services and programs. All of these things can help reduce poverty.

Comparative analysis

The t- ${\mathbb{IFG}}$ is an improved variant of the intuitionistic fuzzy graph that includes an additional parameter referred to as $t,t \in \left[ {0,1} \right].$ By adjusting the ‘ t ’ parameter, uncertainty modeling can be fine-tuned to fit specific requirements and domain characteristics better. By changing the value of the parameter $t$ various decision-making or preference scenarios can be depicted, providing a more precise representation of uncertainty and vagueness. The t- ${\mathbb{IFG}}$ offer various applications in diverse situations and decision-making processes. Their adjustable parameter ‘ t ’ within the closed unit interval enables the capture of varying degrees of conservatism or optimism, allowing for customization according to specific requirements. This notion is beneficial in problem-solving domains where multiple levels of uncertainty, hesitancy, and decision preferences must be considered simultaneously. Their effectiveness shines in complex decision-making scenarios, including medical diagnosis, pattern recognition, and decision support systems, as they can accommodate different levels of uncertainty and hesitancy. The exceptional flexibility and adaptability of t- ${\mathbb{IFG}}$ make them the preferred choice when a more precise representation of uncertainty is necessary.

Furthermore, when the parameter $t$ is assigned a value of $0.1$ within the framework of utilizing t-intuitionistic fuzzy sets to tackle the problem of poverty reduction, it signifies a prudent and somewhat negative assessment of the effectiveness of different factors in alleviating poverty. A membership degree of $0.1$ indicates a weak association between the variables, implying that the impact of poverty reduction is limited. On the other hand, a non-membership degree of $0.9$ signifies a perceived lack of a robust correlation or a fragile link between these variables and the mitigation of poverty. When the degrees of membership and non-membership stay consistent, the elements under examination possess a uniform and equivalent amount of association with a certain factor and a consistent level of non-association. The observed uniformity indicates that all aspects are seen as equally connected to the factor in question, without any noticeable differentiation based on their levels of membership or non-membership. The constancy of ambiguity or reluctance in associating these elements with the factor persists uniformly across all dimensions. Choosing a parameter value of 't' near zero signifies a need for more precision about the impacts on poverty alleviation.

In this research, the concept of t-intuitionistic fuzzy graphs (t- ${\mathbb{IFG}}$ ) has been initiated, and various fundamental features of this phenomenon have been explored. Many set-theoretical operations of t- ${\mathbb{IFG}}$ have been studied, and graphical representations of these operations have been demonstrated. Additionally, the idea of a complement of t- ${\mathbb{IFG}}$ has been defined, and some of its key features have been investigated. The notions of homomorphisms and isomorphisms of t- ${\mathbb{IFG}}$ have been introduced. Furthermore, a practical application of the newly defined technique in reducing poverty has been presented.

The use of t- ${\mathbb{IFG}}$ effectively addresses real-world problems and improves decision-making processes. It is a flexible and robust framework that deals with imprecision and uncertainty in decision-making while optimizing complex systems, recognizing patterns, and offering various applications for computational intelligence. This idea has the potential for future use in healthcare systems, transportation networks, pattern recognition, and machine learning.

Selecting a parameter value 't' close to zero indicates a lack of identifiable specificity in the effects of poverty reduction. In contrast, when the parameter value 't' approaches 1, it strongly signifies a robust and visible correlation with achieving objectives related to reducing poverty. In t- ${\mathbb{IFG}}$ , the parameter 't' measures the level of assurance or uncertainty over the effectiveness of poverty reduction efforts. The extremes of this parameter indicate either a negligible impact or a strong correlation with the desired outcome. Utilizing this calibrated parameter allows decision-makers to precisely adjust the depiction of uncertainty and its influence on analytical results, leading to a sophisticated and flexible structure for tackling the intricate complications of poverty reduction.

One of our primary goals for future studies is to apply the proposed strategy to solve MCDM problems, specifically supplier selection, risk management, and renewable energy selection. The proposed techniques will also be applied to neural networks, clustering, feature selection, and risk management. In addition, some advanced decision-making techniques of complex spherical fuzzy Aczel Alsina aggregation operators 61 will also be studied within the context of the strategies presented in this article.

Data availability

All data generated or analyzed during this study are included in this published article.

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Acknowledgements

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. 4376].

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Department of Basic Sciences, Deanship of Preparatory Year, King Faisal University Al Ahsa, 31982, Al Hofuf, Saudi Arabia

Asima Razzaque & Saima Noor

Department of Mathematics, College of Science, Jazan University, 45142, Jazan, Saudi Arabia

Ibtisam Masmali

Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan

Laila Latif & Umer Shuaib

Department of Mathematics, Division of Science and Technology, University of Education, Lahore, 54770, Pakistan

Abdul Razaq

Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), 11564, Riyadh, Saudi Arabia

Ghaliah Alhamzi

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Razzaque, A., Masmali, I., Latif, L. et al. On t-intuitionistic fuzzy graphs: a comprehensive analysis and application in poverty reduction. Sci Rep 13 , 17027 (2023). https://doi.org/10.1038/s41598-023-43922-0

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Fuzzy covering problem of fuzzy graphs and its application to investigate the Indian economy in new normal

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Anushree Bhattacharya 1 &
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In this paper, new concepts of covering of fuzzy graphs are introduced. The definitions of fuzzy covering maps and fuzzy covering graphs of fuzzy graphs are given. Some special types of fuzzy covering graphs are discussed. Some important theorems to find out the fuzzy covering maps as well as fuzzy covering graphs for different types of fuzzy graphs are described. On the basis of the present situation during the current pandemic COVID-19, the world economic status is highly disruptive and deeply affected by the lock-down process. So, this topic is catching the eye for being an application part of this paper. Also, there are globally seventeen goals to sustain our development, which contain the eighth goal ‘Decent Work and Economic Growth’ having effect on the economy of the world. For this reason, the eighth sustainable development goal is combined with the economic impact of the pandemic for the discussion in the application part. Then some strategies are made to overcome this condition in a better way and perform all the steps to get a better situation in near future.

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1 Introduction

In 2015, the sustainable development goals (SDGs) are set the global development agenda until 2030. The SDGs are universal plans for all countries to end poverty, protect the planet and ensure prosperity for all. There are 17 goals which include 169 targets. They are intended to be tackled as a group rather than individually, the 17 goals are interlinked. The 17 SDGs are:

(1) No Poverty, (2) Zero Hunger, (3) Good Health and Well Being, (4) Quality Education, (5) Gender Equality, (6) Clean Water and Sanitation, (7) Affordable and Clean Energy, (8) Decent Work and Economic Growth, (9) Industry, Innovation, and Infrastructure, (10) Reduced Inequalities, (11) Sustainable Cities and Communities, (12) Responsible Construction and Production, (13) Climate Action, (14) Life Below Water, (15) Life on Land, (16) Peace, Justice, and Strong Institutions, (17) Partnerships and Goals. These SDGs are discussed in more detail in [ 36 , 44 ].

The Eighth SDG is “Decent Work and Economic Growth”, the index-score value of which is connected to the economic status of any country. Many developing countries in the world, perform as much as possible to achieve the target of the Eighth SDG by taking many necessary treads for this purpose. But, in the last few months all the strategies are becoming meaningless and the backbone of the development is broken due to a huge effect of the current pandemic COVID-19.

1.1 Impact of COVID-19 in the world economy

COVID-19 has triggered the deepest global recession in decades. The global economic outlook during the COVID-19 pandemic is a changed world. The COVID-19 pandemic has spread with alarming speed, infecting millions and bringing economic activity to a near-standstill as countries imposed tight restrictions on movement to halt the spread of the virus. The baseline forecast envisions a 5.2% contraction in global GDP in 2020. The pandemic is expected to plunge most countries into recession in 2020, with per capita income contracting in the largest fraction of countries globally since 1870. The pandemic and efforts to contain it have triggered an unprecedented collapse in oil demand and a crash in oil prices. Should COVID-19 outbreaks persist, should restrictions on movement be extended or reintroduced, or should disruptions to economic activity be prolonged, the recession could be deeper. Under this downside scenario, global growth could shrink by almost 8 percent in 2020. While the ultimate outcome is still uncertain, the pandemic will result in contractions across the vast majority of emerging markets and economies development.

1.2 Motivation of the work

To handle real-life all uncertainties, fuzzy mathematics is the best of all fields of mathematics. Specially, the fuzzy graph is one of the strong tools to model the relationship between various features with impreciseness. For the first time, the concept of fuzzy sets is given by Zadeh [ 50 ] and fuzzy graphs are introduced by Rosenfield [ 34 ]. Yager [ 49 ] has described knowledge-based defuzzification. Also, facility location problems are very essential for modeling real-life problems. The idea of covering fuzzy graphs will make the work smooth to analyze a network and console the demand of any fuzzy system. As, the economy is the most important part of our life, which has a markable damage during this pandemic and it will deeply affect our daily-life; so this topic is of much importance to analyze and make necessary strategies to overcome this situation. Fuzzy covering map and fuzzy covering graph of a fuzzy graph are totally new concepts in the area of covering concepts of fuzzy graphs.

In this paper, the new definition of fuzzy covering maps and graphs of fuzzy graphs are introduced. To mix the taste of facility location concept in this context for covering maps and graphs of fuzzy graphs, the vertices of a fuzzy graph are considered as facilities as well as demand points depending on the considered situation of the problem.

1.3 Literature review

There are many works in the field of covering of fuzzy graphs. In the area of vertex covering, Hastad [ 17 ] showed that approximating vertex cover within constant factors less than 7/6 is NP-hard. The factor of Hastad to 1.36 is improved by Dinur and Safra [ 13 ]. The randomness in the problem and the probability theory to minimum weight edge covering problem are discussed by Ni [ 26 ]. On the other side, the credibility theory to find the fuzzy minimum weight edge cover in a fuzzy graph was discussed by him in 2008 [ 27 ]. The fuzzy graphs have been used by Koczy in evolution and optimization of networks [ 18 ]. Pramanik and Pal [ 30 ] have introduced a fuzzy $\phi $ -tolerance competition graph. Samanta and Pal [ 42 ] have proved results on k -competition and p -competition fuzzy graphs, they also studied with fuzzy tolerance graphs [ 41 ] and fuzzy planar graphs [ 43 ]. Rashmanlou et al. [ 33 ] have studied bipolar fuzzy graphs. Interval-valued fuzzy planar graphs are nicely discussed by Pramanik et al. [ 29 ]. In [ 39 ], Samanta et al. described new concepts of fuzzy planar graphs. Samanta et al. [ 38 ] also worked on some results on m -step fuzzy competition graphs. Rashmanlou and Pal [ 32 ] have provided notable theories and important results about the isometry on interval-valued fuzzy graphs. Sahoo and Pal [ 37 ] have done wonderful study on intuitionistic fuzzy competition graphs. Samanta et al. [ 40 ] have developed the methodology of fuzzy colouring of fuzzy graphs. Ghorai and Pal [ 15 ] have worked on m -polar fuzzy planar graphs, they also have given theories on, “Faces and dual of m -polar fuzzy planar graphs” [ 16 ]. An unified approach to fuzzy graph problems is given by Blue et al. [ 9 ]. Chang and Zadeh [ 11 ] have worked on fuzzy mappings. Conditional covering problem is considered by Chaudhry [ 12 ]. Wieslaw and Talebi [ 48 ] have given idea about operations on level graph of bipolar fuzzy graphs. Cayley fuzzy graph on fuzzy group is discussed by Talebi [ 45 ]. Borzooei et al. [ 10 ] have worked on regularity of vague graphs. Also, Talebi et al. [ 46 ] have introduced some new concepts of m -polar interval-valued intuitionistic fuzzy graph.

The definition of neutrosophic graphs was introduced by Akram and Shahzadi [ 5 ]. Mahapatra et al. [ 21 ] has given applications of edge colouring of fuzzy graphs. Some operations on single-valued NGs are given by Akram and Shahzadi [ 4 ]. Mahapatra et al. [ 20 ] have given the concept of radio fuzzy graphs. Sitara and Akram [ 6 ] have shown novel applications in decision-making of neutrosophic graphs. Based on bipolar neutrosophic directed hypergraphs, a new decision-making method is introduced by Akram and Luqman [ 3 ]. In addition, Naz and Akram [ 25 ] have worked on energy in single-valued neutrosophic graphs. Akram [ 2 ] elaborately discussed single-valued neutrosophic graphs. A new way of link prediction in social networks is introduced by Mahapatra et al. [ 19 ]. The concept of generalized neutrosophic planar graphs are described by Mahapatra et al. [ 22 ]. Link prediction in social networks by neutrosophic graphs are studied by Mahapatra et al. [ 23 ]. Bhattacharya and Pal [ 7 ] have worked with vertex covering problems of fuzzy graphs and given application in CCTV installation. Also, clique covering problems are used in optimization by Bhattacharya and Pal [ 8 ]. Pal et al. [ 28 ] have discussed modern trends in fuzzy graphs and their applicability. Dubois and Prade [ 14 ] have discussed about different operations of fuzzy numbers in their studies.

1.4 Novelties

In literature, there are several publications on covering of crisp graphs, but there is a huge scope to work with covering of fuzzy graphs. Also, as per best of our knowledge, there is no work in fuzzy covering maps and graphs of fuzzy graphs. In this paper, some important definitions, works and their applications are provided. The summarized form of the work done are as follows:

Defining fuzzy covering map and fuzzy covering graph of a fuzzy graph.

Grading the vertices of fuzzy graphs with respect to the vertex membership function.

Fuzzy covering map and covering graph for different fuzzy graphs like picture fuzzy graphs, fuzzy perfect graphs and m -polar fuzzy graphs are emphasised.

Some useful notations and abbreviations are given in the Table 1 .

1.5 Organization of the paper

The paper is organised as given. In Sect. 2 , some basic and important definitions are provided. The problem under consideration is described in Sect. 3 . An algorithm to find a fuzzy covering graph of a fuzzy graph is described in Sect. 4 . In Sect. 5 , some important theorems are given for different fuzzy graphs. The fuzzy covering maps and fuzzy covering graphs for picture fuzzy graphs and neutrosophic fuzzy graphs in Sects. 6 and 7 respectively. In Sect. 8 , an application on the impact of COVID-19 in the economy of India with detailed analysis is provided. An entire conclusion is given in Sect. 9 .

2 Preliminaries

Definition 1.

[ 24 ] A fuzzy graph $G=(V,\sigma ,\mu )$ with the underlying crisp graph $G^*=(V,E)$ is a graph defined by the couple of fuzzy membership functions, $\sigma :V \rightarrow [0,1]$ , $\mu :V\times V \rightarrow [0,1]$ where for all $u, v \in V$ , every edges in $G=(V,\sigma , \mu )$ fulfils the condition, $\mu (u,v)\le \sigma (u) \wedge \sigma (v)$ .

The idea of covering in fuzzy graphs was presented by Somasundaram. The author also defined node covering and arc covering in fuzzy graphs using effective arcs and scalar cardinality.

Definition 2

[ 47 ] The neighbourhood of a vertex u of a fuzzy graph $G=(V,\sigma ,\mu )$ is defined by the set,

and, $N[u]=\{u\}\cup N(u)$ is called the closed neighbourhood of u .

Definition 3

[ 47 ] The term $\delta _N(u)=\sum _{v\in N(u)} \sigma (v)$ is called the neighbourhood value of a vertex u for a fuzzy graph $G=(V,\sigma ,\mu )$ .

Definition 4

The minimum neighbourhood value of $G=(V,\sigma ,\mu )$ is,

and the maximum neighbourhood value of $G=(V,\sigma ,\mu )$ is,

Definition 5

[ 35 ] Let, $G=(V_1,E_1)$ and $C=(V_2,E_2)$ be two fuzzy graphs and let $f:V_2 \rightarrow V_1$ be a surjection. Then f is a covering map from C to G if for each $v \in V_2$ , the restriction of f to the neighbourhood of v is a bijection onto the neighbourhood of f ( v ) in G .

If there exists a covering map from C to G , then C is a covering graph, or a lift of G .

We consider a fuzzy graph and using the crisp covering map, we find out the crisp covering graph of the fuzzy graph shown in the Fig. 1 .

Covering graph of a fuzzy graph by crisp covering map

In the Fig. 1 , H is the covering map of the fuzzy graph G by considering a crisp covering map from the vertex set of H to the vertex set of G . In Fig. 1 , the green-coloured vertices of the graph G are considered as facility nodes. Here, the covering radius of G is considered as 2. That is, we can repeat the facility nodes which are situated within 2-edge distances from any vertex present in the fuzzy graph. Also, the edge-membership values remain the same as in the original fuzzy graph by considering the crisp covering map between the vertex sets of G and H . In H , the repeated facility nodes are the sky-coloured vertices.

Definition 6

The equality function on $(V\times V)$ for a fuzzy graph $G=(V, \sigma , \mu )$ to define the fuzzy covering map is denoted by $E:V\times V \rightarrow [0,1]$ and given by,

Also, $E(u,v)\le 1$ implies $\sigma (u)\le \sigma (v)$ for $u,v \in V$ .

Definition 7

Let, $G_1=(V_1, \sigma _1, \mu _1)$ and $G_2=(V_2, \sigma _2, \mu _2)$ be two connected fuzzy graphs.

For two vertex sets $V_1$ and $V_2$ , let $\tilde{f} : V_2 \rightarrow V_1$ be a fuzzy covering map of $G_1$ on $G_2$ with respect to a equality function E on both $V_1$ and $V_2$ , then

$\chi _{\tilde{f}}$ is surjective fuzzy function i.e., $\forall v \in V_2$ , there exists $u\in V_1$ such that $\chi _{\tilde{f}}(u,v)>0$ .

$\chi _{\tilde{f}}$ is injective fuzzy function i.e.,

$\forall u,v \in V_1$ and $\forall z,w \in V_2$ ;

where, $\chi _{\tilde{f}}$ is the membership function of $\tilde{f}$ .

If the above hold, $G_1=(V_1, \sigma _1, \mu _1)$ is called a fuzzy covering graph of $G_2=(V_2, \sigma _2, \mu _2)$ .

Clearly, a fuzzy graph $G=(V, \sigma , \mu )$ be a fuzzy covering graph of itself. In this case, this fuzzy graph is called a fuzzy self-covering graph.

Also, the fuzzy covering map and fuzzy covering graph for a fuzzy graph is not unique.

Let us consider a fuzzy graph and using the definition of fuzzy covering map, we have to find out the fuzzy covering graph of the considered fuzzy graph. The considered graph is named as G , whereas the fuzzy covering graph is denoted by H . Now, we add the facility vertices of the given graph by the help of a fuzzy covering map to construct the graph H . Here, the neighbouring properties of the vertices of the fuzzy graph are conserved and the covering map satisfied all the properties of bijective fuzzy function. In H , we repeat all the facilities of G to cover the fuzzy graph G by splitting the vertices at $v_4$ and $v_5$ . As shown in Fig. 2 , the facility vertex $v_4$ is adjacent vertex of the vertices $v_5$ and $v_1$ ; also the facility vertex $v_5$ is adjacent to the vertices $v_4$ and $v_1$ . This neighbourhood property of the vertices $v_4$ and $v_5$ are kept in the fuzzy covering map H of fuzzy graph G .

Fuzzy covering graph of a fuzzy graph

In the Fig. 2 , the fuzzy graph H is a fuzzy covering map of the fuzzy graph G by considering the fuzzy covering map $\tilde{f}$ which have the membership function, $\chi _{\tilde{f}}$ is given by

The above defined membership function is a fuzzy surjective as well as fuzzy injective function, therefore H is a fuzzy covering graph of the given fuzzy graph G .

In H , yellow-coloured vertices are the same as in the original graph G and the green-coloured vertices are the repeated facility nodes to construct the fuzzy covering graph H of the fuzzy graph G .

Definition 8

Let, $G_1=(V_1, \sigma _1, \mu _1)$ and $G_2=(V_2, \sigma _2, \mu _2)$ be two fuzzy graphs.

For two vertex sets $V_1$ and $V_2$ , let $\tilde{f} : V_2 \rightarrow V_1$ be a strong fuzzy covering map of $G_1$ on $G_2$ with respect to a equality function E on both $V_1$ and $V_2$ , then

$\chi _{\tilde{f}}$ is strong surjective fuzzy function i.e., $\forall v \in V_2$ , there exists $u\in V_1$ such that

$\forall u,v \in V_1$ and $\forall z,w \in V_2$ .

If the above hold, $G_1=(V_1, \sigma _1, \mu _1)$ is called a strong fuzzy covering graph of $G_2=(V_2, \sigma _2, \mu _2)$ .

Definition 9

[ 31 ] Total degree of a vertex u of a fuzzy graph $G=(V, \sigma , \mu )$ is defined as,

Definition 10

Index of a fuzzy graph $G=(V, \sigma , \mu )$ is defined as,

And, the term ‘Covering index’ is used for the index of the covering graph of a fuzzy graph and it is denoted by $C_I(G)$ .

As finite numbers of vertices present in the vertex-sets, so the number of fuzzy covering maps from one fuzzy graph to another fuzzy graph is finite always and countable.

Fuzzy covering map is clearly a fuzzy isomorphism. That is, $G_1$ is homeomorphic to $G_2$ .

Also, a fuzzy covering map is a fuzzy homomorphism.

Definition 11

For any graph G , it is possible to construct the bipartite double cover of G , which is a bipartite graph and a double cover of G . The bipartite double cover of G is the tensor product of graphs $G \times K_2$ (Fig. 3 ).

Double covering of a fuzzy graph

If G is already bipartite, its bipartite double cover consists of two disjoint copies of G . A graph may have many different double covers other than the bipartite double cover.

Definition 12

The vertices of a fuzzy graph $G=(V, \sigma , \mu )$ are graded with respect to the vertex membership function as follows:

$\sigma (u)=1$ implies, u is called the strongest vertex, for $u \in V$ .

$\sigma (u) \in (0.7, 1)$ implies, u is called first grade vertex, for $u \in V$ .

$\sigma (u) \in (0.5, 0.7)$ implies, u is called second grade vertex, for $u \in V$ .

$\sigma (u) \in (0, 0.5)$ implies, u is called third grade vertex, for $u \in V$ .

Definition 13

[ 47 ] Let, G be the composition $G_1[G_2]$ of graph $G_1$ with $G_2$ . Let $(\sigma _i, \mu _i)$ be a fuzzy subgraph of $G_i$ , for $i=1,2$ . Then $(\sigma _1 \circ \sigma _2, \mu _1 \circ \mu _2)$ is a fuzzy subgraph of $G_1[G_2]$ , where

The fuzzy graph $(\sigma _1 \circ \sigma _2, \mu _1 \circ \mu _2)$ is called the composition of $(\sigma _1, \mu _1)$ with $(\sigma _2, \mu _2)$ .

Definition 14

[ 1 ] Let $G^*=(V,E)$ be a graph. A pair $G=(A, B)$ is called a picture fuzzy graph on $G^*$ where $A=(\mu _A, \eta _A, \nu _A)$ is a picture fuzzy set on V and $B=(\mu _B, \eta _B, \nu _B)$ is a picture fuzzy set on $E\subset V\times V$ such that for each arc $uv \in E$ ,

Definition 15

[ 15 ] An m -polar fuzzy graph is a pair $G=(C, D)$ , where $C:V \rightarrow [0,1]^m$ is an m -polar fuzzy set in V and $D:V \times V \rightarrow [0,1]^m$ is an m -polar fuzzy relation on V such that $p_i \circ D(xy) \le inf \{p_i \circ C(x), p_i \circ C(y)\}$ for all $x, y \in V$ .

Definition 16

A fuzzy graph $G=(V, \sigma , \mu )$ on ( V , E ) is called a perfect fuzzy graph if $\sigma (v)=1$ , for all $v \in V$ .

3 Problem under consideration

Consider a connected, undirected, unweighted fuzzy graph $G=(V, \sigma , \mu )$ with the following facility location problem as a covering problem of the fuzzy graph.

Let, the fuzzy graph $G=(V, \sigma , \mu )$ be given and some facilities ( S ( R )) are already located with covering radius say R . Now, introduced some more facilities $(S(R'))$ in G with given covering radius $R(\le R')$ , such that the cost of $(S(R)\cup S(R'))$ is maximum. In this paper, we consider the covering radius as a length in a fuzzy graph in terms of the number of edges between two vertices of that fuzzy system.

This problem can be referred to as upgradation of the network.

The above problem can be solved by the flavour of fuzzy covering map and fuzzy covering graph of a fuzzy graph. Since, fuzzy covering map is an edge-preserving bijection from one fuzzy graph to another fuzzy graph; therefore, the given fuzzy system will be covered by adding some existing facilities which are more relevant for maximum coverage and maximum cost of the fuzzy network, also they are situated within a certain covering radius of the fuzzy graph. This locating process of the facilities in the fuzzy graph is determined by the definition of fuzzy covering map.

In this covering concept, we consider the cost of covering for a fuzzy covering graph of a fuzzy graph as the covering index of that fuzzy graph throughout the paper. The main aim is to maximize the cost of covering by repeating a minimum number of facilities.

4 Algorithm to construct fuzzy covering graph of a fuzzy graph

5 Some theorems

In this portion, some important theorems on fuzzy covering maps and fuzzy covering graphs are described for different types of fuzzy graphs. Also, the theorems are discussed by considering some relevant examples.

Let, $G_1=(V_1, \sigma _1, \mu _1)$ and $G_2=(V_2, \sigma _2, \mu _2)$ be two fuzzy graphs. If $G_1 \circ G_2$ is the composition graph of the two fuzzy graphs $G_1$ and $G_2$ , then $C_I(G_1)+C_I(G_2) \le C_I(G_1 \circ G_2)$ . In the other words, the fuzzy covering graph of the composition graph of two fuzzy graphs has more coverage than the total coverage by the fuzzy covering graphs of two individual fuzzy graphs.

Now, consider two fuzzy graphs $G_1$ and $G_2$ given in the Fig. 4 . By the definition of fuzzy covering map and fuzzy covering graph of fuzzy graphs, clearly any fuzzy graph can be a fuzzy covering graph of it’s own. Therefore, $H_1$ and $H_2$ are the fuzzy covering graphs of the fuzzy graphs $G_1$ and $G_2$ respectively, where $H_1$ has the same structure and behaviour like $G_1$ and $H_2$ has the said same as $G_2$ . The fuzzy covering graphs are also shown in Fig. 4 .

Fuzzy covering graphs of two fuzzy graphs

Therefore, the covering index of the fuzzy graph $G_1$ is given by:

and, the covering index of the fuzzy graph $G_2$ is given by:

The composition graph of the two fuzzy graphs $G_1$ and $G_2$ is determined by the help of the definition composition of two fuzzy graphs given in the preliminaries section. The composition graph $G_1 \circ G_2$ is shown in the Fig. 5 . The facility vertices of the graph $G_1 \circ G_2$ are violet coloured nodes.

Here, we consider the trivial case of fuzzy covering graphs of $G_1$ and $G_2$ i.e., the self-covering graphs shown as in $H_1$ and $H_2$ respectively. In $H_1$ , all the yellow-coloured vertices remain the same as facility nodes to cover $G_1$ and in $H_2$ , the sky-coloured vertices of $G_2$ remain the same as facilities to cover $G_2$ .

Composition of given two fuzzy graphs

Now, we have to find out the fuzzy covering graph of the composition graph $G_1 \circ G_2$ . For this purpose, consider the fuzzy covering map $\tilde{f}$ where the membership function of $\tilde{f}$ is, $\chi _{\tilde{f}}(u,v)=\{\sigma (u) \wedge \sigma (v)\}min\big \{\{E(u,u_i): u_i \in N(u)\} \wedge \{E(v,v_i):v_i \in N(v)\}\big \}$ . In the fuzzy covering map, there are more facilities than the given fuzzy graph to cover up the fuzzy system; then there will also be more edges. For this purpose, we assume that the vertex membership values for the facility nodes remain unchanged. The edge-membership function is redefined for those new edges with the help of the equality function as follows:

By the above considerations, we get the fuzzy covering graph of the composition graph of the given two fuzzy graphs is given in the Fig. 6 . In G , the yellow-coloured vertices are the repeated facilities of $G_1 \circ G_2$ to get maximum coverage with maximum covering index.

Fuzzy covering graph of the composition graph

Now, the covering index of $(G_1 \circ G_2)=$ Index of G $=(0.3+1)+(0.3+0.95)+(0.2+0.4)+(0.3+0.3)+(0.2+0.4)+(0.4+0.25)+(0.2+0.4)+(0.3+0.95)+(0.4+0.25)+(0.3+1)+(0.2+0.4)+(0.3+0.3)=10$ .

Therefore, $C_I(G_1)+C_I(G_2) \le C_I(G_1 \circ G_2)$ (proved).

Let us consider an m -polar fuzzy graph, a pair $G=(C, D)$ , where $C:V \rightarrow [0,1]^m$ is an m -polar fuzzy set in V and $D:V \times V \rightarrow [0,1]^m$ is an m -polar fuzzy relation on V such that $p_i \circ D(xy) \le inf \{p_i \circ C(x), p_i \circ C(y)\}$ for all $x, y \in V$ . Then there is a fuzzy covering map whose membership function is given by: $\chi _{\tilde{f}}(u,v)=min\{p_i \circ C(u), p_i \circ C(v)\}$ for $u, v \in V$ . That is, a m -polar fuzzy covering graph of a m -polar fuzzy graph by considering some additional vertices which are the facility nodes of the fuzzy graph.

Let us consider a m -polar fuzzy graph G given by the Fig. 7 . As in the statement of the above theorem, considering the same fuzzy covering map together with it’s membership function; we can find out an m -polar fuzzy graph as a fuzzy covering graph of the fuzzy graph G .

Considered m -polar fuzzy graph

By adding some more existing facilities of G in a new fuzzy graph H , we can determine the m -polar fuzzy covering graph. For this purpose, we define the edge-membership function to evaluate the edge-membership values of the new edges in the fuzzy covering graph as follows:

In Fig. 10 , the pink-coloured vertices are the facilities of the fuzzy graph G and let, the covering radius is 2. The fuzzy covering graph H of the fuzzy graph G is obtained by repeating the facilities $v_1, v_2, v_3$ which are red-coloured in H and the new edges have their membership values determined by the help of the definition of fuzzy covering map and equality function. Then, using the considerations, we have the fuzzy covering graph H which is also an m -polar fuzzy graph is given in the Fig. 8 .

Fuzzy covering graph of the m -polar fuzzy graph

The fuzzy covering graph of a fuzzy perfect graph is always a crisp graph and the fuzzy covering map is always a strong fuzzy covering map for a fuzzy perfect graph.

Let, $G=(V, \sigma , \mu )$ on ( V , E ) be a perfect fuzzy graph if $\sigma (v)=1$ , for all $v \in V$ . Consider a fuzzy covering map $\tilde{f}$ whose membership function $\chi _{\tilde{f}}$ is given by $\chi _{\tilde{f}}(u,v)=\{\sigma (u) \wedge \sigma (v)\}min\big \{\{E(u,u_i): u_i \in N(u)\} \wedge \{E(v,v_i):v_i \in N(v)\}\big \}$ . Then, $\chi _{\tilde{f}}(u,v)=1 \forall u, v \in V$ . We consider this membership function as the edge-membership function of the new additional edges in the fuzzy covering graph of G .

Let us consider a fuzzy perfect graph G given in the Fig. 9 . As in the statement of the above theorem, we consider the same fuzzy covering map with membership function and same edge-membership function, the fuzzy covering graph H is obtained. This graph H is obviously a crisp graph given in the Fig. 9 . Clearly, there exists a fuzzy bijective function between the vertex sets of G and H ; here the corresponding vertices of G and H are same coloured which are evaluated by the help of the fuzzy covering map.

Fuzzy covering graph of a perfect fuzzy graph

6 Fuzzy covering maps for picture fuzzy graph

The picture fuzzy graph is a particular type of fuzzy graph which includes more conditions for its vertex and edge membership functions. Picture fuzzy graphs are more relevant to use in any real-life fuzzy environment for mathematical modelling in graph theoretic approach. For this reason, the construction of the fuzzy covering graph of a picture fuzzy graph is more important than any other general type fuzzy graph. It will be very helpful in the decision-making process by the help of fuzzy covering technique of picture fuzzy graphs.

Let $G^*=(V,E)$ be a graph. A pair $G=(A, B)$ is called a picture fuzzy graph on $G^*$ where $A=(\mu _A, \eta _A, \nu _A)$ is a picture fuzzy set on V and $B=(\mu _B, \eta _B, \nu _B)$ is a picture fuzzy set on $E\subset V\times V$ such that for each arc $uv \in E$ ,

Let us consider a new fuzzy graph, $H^*=(V_2, E_2)$ with a pair $H=(C,D)$ is also a picture fuzzy graph. Here $C=(\mu _C, \eta _C, \nu _C)$ is a picture fuzzy set on $V_2$ and $D=(\mu _D, \eta _D, \nu _D)$ is a picture fuzzy set on $E\subset V_2\times V_2$ such that for each arc $xy \in E_2$ ,

where, E is the equality function.

Then, we have a fuzzy covering map $\tilde{f}:V_2 \rightarrow V_1$ whose membership function is given by:

Therefore, H is a fuzzy covering graph of G i.e., the fuzzy covering graph of a picture fuzzy graph is also a picture fuzzy graph.

Let us consider a picture fuzzy graph $G=(A,B)$ given in the Fig. 10 , where $A=(\mu _A, \eta _A, \nu _A)$ is a picture fuzzy set on V and $B=(\mu _B, \eta _B, \nu _B)$ is a picture fuzzy set on $E\subset V\times V$ such that for each arc $uv \in E$ ,

$\mu _B(u,v)\le min (\mu _A(u), \mu _A(v))$ , $\eta _B(u,v)\le min (\eta _A(u), \eta _A(v))$ , $\nu _B(u,v)\ge max (\nu _A(u), \nu _A(v))$ . We have to find out a fuzzy covering map together with a fuzzy covering graph of the considered picture fuzzy graph.

Considered picture fuzzy graph

Assuming a fuzzy covering map $\tilde{f}:V(H) \rightarrow V(G)$ whose membership function is given by:

Also, the edge-membership function for the new edges in the fuzzy covering graph is given by:

By the above consideration, we find out the fuzzy covering graph $H=(C, D)$ of the picture fuzzy graph $G=(A,B)$ . The graph H is also a picture fuzzy graph and given by the Fig. 11 , where the yellow-coloured vertices are the existing facilities in the given picture fuzzy graph G and we add the more necessary facilities among them to construct the fuzzy covering graph H . The new added facilities are violet-coloured vertices in H .

Fuzzy covering graph of the picture fuzzy graph

7 Fuzzy covering graphs of neutrosophic fuzzy graphs

Neutrosophic fuzzy graphs are another special type of fuzzy graph which have three types of membership functions of any vertex for any result in the decision-making process. In many situations, we have to conclude a decision among three choices: truth, indeterminacy and falsity. If this particular situation needs any covering with the placements of the existing facilities in a fuzzy network within a given covering radius, then the fuzzy covering graph of neutrosophic fuzzy graphs have the most impact for maximum coverage for this purpose. In this part, the covering graph of a neutrosophic fuzzy graph is to be determined.

A neutrosophic fuzzy graph (NF-graph) with underlying set V is defined to be a pair $N_G=(A,B)$ where,

1. The functions $T_A:V \rightarrow [0,1], I_A:V \rightarrow [0,1]$ and $F_A:V \rightarrow [0,1]$ denote the degree of truth-membership, degree of indeterminacy-membership and falsity-membership of the element $v_i \in V$ , respectively, and

2. $E \subset V \times V$ where, functions $T_B:V\times V \rightarrow [0,1]$ , $I_B:V\times V \rightarrow [0,1]$ and $F_B:V\times V \rightarrow [0,1]$ are defined by,

for all $v_i, v_j \in V$ where . means the ordinary multiplication; denote the degree of truth-membership, indeterminacy-membership and falsi-membership of the edge $(v_i,v_j)\in E$ respectively, where

for all $(v_i,v_j)\in E$ for $(i,j=1,2,\ldots ,n)$ .

We call ‘ A ’ the neutrosophic fuzzy vertex set of V , ‘ B ’ the neutrosophic fuzzy edge set of E , respectively.

Let us consider a neutrosophic fuzzy graph G given by the Fig. 12 . We have to find a fuzzy covering graph of the neutrosophic fuzzy graph G whose facility nodes are orange-coloured vertices.

Considered neutrosophic fuzzy graph G

Let, H be the fuzzy covering graph of the fuzzy graph G . Assuming a fuzzy covering map $\tilde{f}: V(H) \rightarrow V(G)$ with the three underlying fuzzy covering maps for the three different degree membership functions of the vertices of the fuzzy graph; we have to construct the graph H . The membership function of the fuzzy covering map for the neutrosophic fuzzy graph is given by,

for all $v_i,v_j \in V$ .

The edge-membership function for the new edges in the fuzzy covering graph is given by,

for all $v_i, v_j \in V(H)$

By this consideration, we construct the graph H , where blue-coloured vertices are the new added facility nodes to cover the graph G . Here, the graph H is also a neutrosophic fuzzy graph which is given by the Fig. 13 .

Fuzzy covering graph of the neutrosophic fuzzy graph G

8 An application

India is one of the developing countries of the world, trying as much as possible to develop the economic situation for achieving the target of eighth SDG “Decent Work and Economic Growth”. But, in the present situation, the main obstacle for fulfilling the target in eighth SDG due to a long-time lock-down process to control the pandemic COVID-19. It is very difficult for a country like India with a wide population and a developing economic status to overcome this critical situation. The economic impact of the 2020 Corona virus pandemic in India has been largely disruptive.

8.1 SDG 8: Global context and position of India

SDG 8 seeks to achieve higher economic productivity and job creation through diversification and innovations in technology, while at the same time protecting labour rights and promoting a safe and secure working environment. It also aims to eradicate forced labour, human trafficking and child labour. Despite gains in human development, narrowing gap of per capita income between high-and-lower-income countries and improvements in labour productivity; only limited success has been achieved globally on most of the SDG 8 targets.

With one person out of every six on the planet living in India, the country has the potential to be the engine of the global economic process. India, with a growth rate of 6.8% in 2018–2019, is recognised as one of the fastest-growing large economies in the world. With 54.3% share of Indian GDP coming from the Services Sector, which continues to record a growth rate of 7–8 percent during each quarter of 2018–2019. India continues to register a high growth rate. The government has taken many steps to ensure further consolidation at the macro-economic level, strengthening of investment sentiments, promotion of entrepreneurship and creation of a skilled workforce. One of the key challenges in India has been the declining participation of female workers in the labour force. The seven factors of SDG 8 are:

Annual growth rate of net Domestic Product (NDP) per capita ( $f_1$ ).

Ease of doing business score ( $f_2$ ).

Unemployment rate( $f_3$ ).

Labour force participation rate ( $f_4$ ).

Number of banking outlets per 1,00,000 population ( $f_5$ ).

Percentage of households with a bank account ( $f_6$ ).

Proportion of women account holders under “Pradhan Mantri Jan Dhan Yojona” ( $f_7$ ).

As per the annual report of Transforming India project 2019–2020, declared on near November, 2019; the position of India in SDG 8 based on the national index score with seven indicators are given by the Table 2 .

8.2 Economic impact of the COVID-19 pandemic in India

Cause: COVID-19 pandemic-induced market instability and lock-down.

Sharp rise in unemployment (45%) (denoted by a ).

Stress on supply chain (53%) (denoted by b ).

Decrease in government income (23.9%) (denoted by c ).

Collapse of the tourism and hospitality industry (70%) (denoted by d ).

Reduced consumer activity (in Stock markets) (13.15%) (denoted by e ).

Plunge in fuel consumption and rise in LPG sales (46%) (denoted by f ).

Fall in trade with China (25%) (denoted by g ).

India’s growth in the fourth quarter of the fiscal year 2020 went down to 3.1% according to the Ministry of Statistics. The Chief Economic Adviser to the Government of India said that this drop is mainly due to the COVID-19 pandemic effect on the Indian economy. State Bank of India research estimates a contraction of over 40% in the GDP in Q1FY21. The contraction will not be uniform, rather it will differ according to the various parameters such as state and sector.

Unemployment rose from 6.7% on 15th March to 26% on 19th April and then back down to pre-lock-down levels by mid-June. During the lock-down, an estimated 14 crore (140 million) people lost employment while salaries were cut for many others. More than 45% of households across the nation have reported an income drop as compared to the previous year. The Indian Economy was expected to lose over Rs. 32,000 Crore (US $ 4.5 billion) every day during the first 21 days of complete lock-down, which was declared following the corona virus outbreak. Under complete lock-down, less than a quarter of India’s $ 2.8 trillion economic movement was functional. Upto 53% of businesses in the country were projected to be significantly affected. Supply chains have been put under stress with the lock-down restrictions in place; initially, there was a lack of clarity in streamlining what an ‘essential’ is and what is not. A large number of farmers around the country who grow perishables also faced uncertainty.

GST is now an important term in Indian economy. GST stands for Goods and Services Tax is an indirect tax(or, consumption tax) used in India on the supply of goods and services. It is a comprehensive multistage, destination-based tax: comprehensive because it has subsumed almost all the indirect taxes except a few state taxes.

8.3 Self-reliant India mission

On 12th May, 2020, the Prime Minister addressed the nation saying that the Corona virus pandemic was an opportunity for India to increase self-reliance. He proposed the “Atmanirbhar Bharat Abhiyan (Self-reliant India Mission)” economic package.

The Finance Minister stated that the aim was to ‘spur growth’ and ‘self-reliance’, adding that ‘self-reliant India’ does not mean cutting off from the rest of the world. The strategy of combining fiscal and monetary, liquidity measures was defended by the government. On 18th April, 2020; India changed its foreign direct investment (FDI) policy to curb ‘opportunistic takeover/acquisitions of Indian Companies due to the current pandemic’, according to Department for Promotion of Industry and Internal Trade. With the fall in global share prices, there is concern that China could take advantage of the situation, leading to hostile takeovers. While the new FDI policy does not restrict markets, the policy ensures that all FDI from countries that share a land border with India will now be under scrutiny of the Minister of Commerce and Industry. However, by August 2020, Chinese exports to India had fallen by 25%. The overall stimulus provided by ‘Atmanirbhar Bharat Package (Rs. Cr.)’ is given by the Fig. 14 .

The overall stimulus provided by Self-reliant India Package (Rs. Cr.)

8.4 Economy recovery of India

Arthur D. Little, an international consulting firm, has advised that India will most probably see a W-shaped recovery. Mythili Bhusnurmath writes in ‘The Economic Times’ that U-shaped recovery is the most likely followed by an L-shaped recovery. CRISIL chief economist says if things go well, that if the virus is contained, we can expect a V-recovery, otherwise it will end as a U-recovery.

On 2nd July, 2020; ‘The Times of India’ reported that a number of economic indicators such as the manufacturer’s purchasing manager’s index, goods movement, GST collections, electricity usage and rail freight transport showed significant improvement as compared to previous months under lock-down. The Reserve Bank of India had said the impact of COVID-19 is more severe than anticipated and the GDP growth during 2020–2021 is likely to remain in the negative territory. It projected some pick-up in growth impulses from the second-half (October–March) of 2020–2021 onwards.

8.5 Fuzzy graph construction

To analyze all the above described situations with uncertainties and impreciseness by the concept of fuzzy graphs and fuzzy covering map along with fuzzy covering graph of that fuzzy graph, we have to construct mainly two fuzzy graphs with the help of available real-life data and information. For this application, the covering radius is taken as 2 for the ease of handling the real-life fuzziness by a fuzzy graph.

The first fuzzy graph is constructed to represent the performance status of India for the eighth SDG ‘Decent Work and Economic Growth’ with the contributions of all states/UTs of India. The national index-score for SDG 8 along with seven different factors of this SDG are considered for evaluating vertex-membership values of the fuzzy graph. For assigning the vertex membership values of the fuzzy graph, the scaling technique by dividing the index-score by 100. Then, we get the index-score values from [0, 100] to a suitable membership value for vertices in [0, 1]. Here, we consider seven vertices for seven different factors and the equality function given in the preliminaries is used to evaluate the edge-membership values of the fuzzy graph. We assume that if any factor directly or indirectly influences another factor of the SDG 8, then there is an edge between the vertices which represent these two factors. The edge-membership function is redefined for those new edges with the help of the equality function as follows:

This fuzzy graph is denoted by $S_1=(V_1, \sigma _1, \mu _1)$ , given by the Fig. 15 .

The second fuzzy graph is developed to represent the economic impact of COVID-19 pandemic in India. In real-life context, we get all the values in increasing or decreasing in all the parameters which have an important effect in the economic status of India, are all in percent value. We scaled all the data by dividing all by 100 to get the values in [0, 1]. These scaled data are subtracted from 1 to get the positive impact of this pandemic as a complement concept. As described above, there are mainly seven areas which have a strong effect during this pandemic with long-time lock-down situations. Here, these seven factors are treated as vertices and having an influence between two vertices, then there is an edge between them. Also, the same equality function given in the preliminaries section is used to assign the edge-membership values of that fuzzy graph. The edge-membership function is redefined for those new edges with the help of the equality function as follows:

This fuzzy graph is denoted by $S_2=(V_2, \sigma _2, \mu _2)$ , given by the Fig. 16 .

Fuzzy graph representation of the performance of India for SDG 8

Fuzzy graph representation of the impact of COVID-19 on Indian economy

8.6 Covering graph for the fuzzy graph with respect to SDG 8 in India

In this portion, the fuzzy covering graph of the fuzzy graph $S_1$ is to be determined by using Algorithm A1. We define a fuzzy covering map $\tilde{f}$ whose membership function, $\chi _{\tilde{f}}$ is given by

As in the fuzzy graph $S_1$ , the vertices represent the effective factors for the eighth sustainable development goal “Decent Work and Economic Growth”. Here $f_1$ i.e., annual growth rate of Net Domestic Product (NDP) per capita, $f_2$ i.e., ease of doing business score and $f_6$ i.e., percentage of households with a bank account are the most positive factors for achieving the national target for SDG 8. Therefore, to find out the fuzzy covering graph of the fuzzy graph $S_1$ , these facility nodes i.e., $f_1$ , $f_2$ and $f_6$ are included in the fuzzy covering graph to get a better coverage in the performance of India for SDG 8. Due to the inclusion of these facilities, some new edges coming out in the fuzzy covering graph. It will be noted that the membership values of the new facilities which are placed in the existing fuzzy system to get the covering graph, remain the same as in the original network. Also, the edge-membership function for the new edges in the fuzzy covering graph are determined by the following function:

Thus, the fuzzy covering graph to get a better coverage for the performance of India for SDG 8 is obtained and it is denoted by $C(S_1)$ , given in the Fig. 17 .

Fuzzy covering graph of the fuzzy graph $S_1$

From Fig. 17 , the new included facility nodes in $C(S_1)$ are coloured by different colours.

Therefore, the covering index for the fuzzy graph representing the status of India for SDG 8 is,

8.7 Covering graph for the fuzzy graph with respect to the economic impact of COVID-19 in India

In this part, the fuzzy covering graph of the fuzzy graph $S_2$ is to be determined with the help of Algorithm A1. Defining a fuzzy covering map $\tilde{f}$ which have the membership function $\chi _{\tilde{f}}$ is given by $\chi _{\tilde{f}}(u,v)=\{\sigma (u) \wedge \sigma (v)\}min\big \{\{E(u,u_i): u_i \in N(u)\} \wedge \{E(v,v_i):v_i \in N(v)\}\big \}$ .

As said before, the impact of the current pandemic COVID-19 in the economy of the world or India is in the negative sense and has broken the backbone of the economic status of every country. Therefore, to get a positive impact of this pandemic, we subtract the negative impact scaled value from 1 and assign this value as the vertex membership value. Among the main outcomes due to this pandemic, if the two most notable outcomes b i.e., stress on supply chain and c i.e., decrease in government income are to be improved immediately to overcome this situation. Therefore, these two facility nodes b and c are to be included to determine the fuzzy covering graph of the fuzzy graph representation for this case. The vertex and edge membership functions are treated the same as the concept for constructing the fuzzy covering graph of the fuzzy graph presenting the performance status of India for SDG 8. That is, the membership values of the new facilities which are placed in the existing fuzzy system to get the covering graph, remain the same as in the original network. Also, the edge-membership function for the new edges in the fuzzy covering graph are determined by the following function:

Thus, the fuzzy covering graph to get a better situation from this status due to the pandemic, is obtained and it is denoted by $C(S_2)$ , given in the Fig. 18 .

Fuzzy covering graph of the fuzzy graph $S_2$

From Fig. 18 , the new included facility nodes in $C(S_2)$ are coloured by different colour.

Therefore, the covering index for the fuzzy graph representing the status of India due to the current pandemic is,

8.8 Composition of two fuzzy self-covering graphs for above two cases

Now, we find out the composition fuzzy graph of the two fuzzy self-covering graphs for performance status of India in SDG 8 and the impact of the current pandemic on Indian economy respectively. Therefore, by using the definition of composition of two fuzzy graphs, the composition of $S_1$ and $S_2$ are performed, which is given by the Fig. 19 .

Composition of two fuzzy self-covering graphs for above two cases

In the Fig. 19 , for complexity of visual capacity, the vertices are denoted by $1,2,3,\ldots ,49$ , also there is no naming and assignation of the vertex and edge membership values. Therefore, the detailed information is given in a tabular form by Table 3 .

By the help of Table 3 , the membership values of all edges present in the Fig. 19 are calculated.

Therefore, the index of the graph $S_1 \circ S_2$ is given by, 72.89.

Also, by Theorem 1 , we can conclude that $(S_1 \circ S_2)_I \ge C_I(S_1)+ C_I(S_2)$ and it is verified by the above computations.

8.9 Analysis of the results by the fuzzy covering graphs

Now, from Table 3 , it can be shown that the vertices 17, 19, 21, 38, 40, 45, 47, 49, 42 of the composition graph are of first category. That is, the topics presented by these vertices to make stable the Indian economy have the most important role. In the other words, the parameters combinations $(f_3,c)$ , $(f_3,e)$ , $(f_3,g)$ , $(f_6,c)$ , $(f_6,e)$ , $(f_7,c)$ , $(f_7,e)$ , $(f_7,g)$ and $(f_6,g)$ are to be sincerely handled by the Indian government to repair the damage of this pandemic situation and to be a better performer for SDG 8.

Now, by the real-valued data, impact of $S_1$ is 64, impact of $S_2$ is 75 respectively. By the constructed fuzzy graph data, the impact of $S_1$ is 18.705, impact of $S_2$ is 19.754 respectively. Also, $C_I(S_1)+C_I(S_2)=38.459$ and $(S_1 \circ S_2)_I = 72.89$ . With these three sets of data, a comparison is shown in the Fig. 20 .

Comparison analysis by three sets of data related to the application

From Fig. 20 , it can be concluded that the defined fuzzy covering map gives a fuzzy covering graph of the given fuzzy graphs for a better coverage by inclusion of new facilities in the existing fuzzy network. Also, the composition of two fuzzy self-covering graphs has a greater impact than two individual fuzzy covering graphs clearly.

8.10 Decision for this application

From the above analysis process and discussion, for this particular application, the following are to be taken immediately:

the unemployment rate ( $f_3$ ) has to be decreased in any way.

percentage of households with a bank account ( $f_6$ ) has to be increased.

women’s of our society must to be empowered i.e., in another way, proportion of women account holders under PMJDY in India ( $f_7$ ) is to be increased.

the total income of the government ( c ) must to be increased by taking all necessary steps.

consumer activity throughout the country ( e ) has to be increased by making the economy liquidity for the common people of India.

dependency in the foreign business is to be reduced ( g ) and take more interest and importance in self-reliant India mission.

If the above are maintained, then the country will come out from this broken economic situation due to the pandemic COVID-19 and would be a better performer in fulfilling the target of eighth sustainable developmental goal ‘Decent Work and Economic Growth’ as a very fast developing country in the world.

9 Conclusion

In this paper, we introduced a new approach of covering problems of fuzzy graphs. The new definitions of fuzzy covering maps and fuzzy covering graphs of a fuzzy graph are introduced. The finding of fuzzy covering graphs of a fuzzy graph are discussed as another form of covering problem to locate new facility points in a existing fuzzy system. In this process, the facility location problem is solved by a new approach of fuzzy covering graphs of fuzzy graphs. Some theorems are showing the actual taste of this new concept of covering of fuzzy graphs. In the application part, a decision-making process based on the fuzzy covering graphs concept is illustrated. Also, a strategy making decision given for a real-life problem to overcome the pandemic situation impact on the Indian economy.

All the given new definitions and theoretical approaches will be applicable for connected, simple and undirected fuzzy graphs only. These definitions and concepts are to be modified for disconnected, directed fuzzy graphs and also other exceptional situations. In future, other types covering like the path-covering, edge-covering concepts are to be generalized by using the proposed definitions in this article. Also, the fuzzy path-covering and fuzzy edge-covering graphs of fuzzy graphs will be more relevant to model facility-location problems in new ways.

In the future work, various types of covering problems of the fuzzy graphs are to be considered to solve many real-life problems. Also, the concept of double covering of a fuzzy graph will be considered in our upcoming work. We are interested to deal with another sustainable developmental goals for a better world.

Availability of data and materials

All the data are collected from ‘Google’.

Code availability

Not applicable.

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Bhattacharya, A., Pal, M. Fuzzy covering problem of fuzzy graphs and its application to investigate the Indian economy in new normal. J. Appl. Math. Comput. 68 , 479–510 (2022). https://doi.org/10.1007/s12190-021-01539-4

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Application of connectivity index of cubic fuzzy graphs for identification of danger zones of tsunami threat

Roles Conceptualization, Supervision

Affiliation Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China

Roles Conceptualization, Data curation, Methodology

* E-mail: [email protected]

Roles Conceptualization, Investigation, Validation, Writing – original draft

Affiliation Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, Pakistan

Roles Investigation, Validation, Writing – original draft, Writing – review & editing

Xiaolong Shi,
Saeed Kosari,
Saira Hameed,
Abdul Ghafar Shah,
Samee Ullah

Published: January 30, 2024
https://doi.org/10.1371/journal.pone.0297197
Reader Comments

Fuzzy graphs are very important when we are trying to understand and study complex systems with uncertain and not exact information. Among different types of fuzzy graphs, cubic fuzzy graphs are special due to their ability to represent the membership degree of both vertices and edges using intervals and fuzzy numbers, respectively. To figure out how things are connected in cubic fuzzy graphs, we need to know about cubic α −strong, cubic β −strong and cubic δ −weak edges. These concepts better help in making decisions, solving problems and analyzing things like transportation, social networks and communication systems. The applicability of connectivity and comprehension of cubic fuzzy graphs have urged us to discuss connectivity in the domain of cubic fuzzy graphs. In this paper, the terms partial cubic α −strong and partial cubic δ −weak edges are introduced for cubic fuzzy graphs. The bounds and exact expression of connectivity index for several cubic fuzzy graphs are estimated. The average connectivity index for cubic fuzzy graphs is also defined and some results pertaining to these concepts are proved in this paper. The results demonstrate that removing some vertices or edges may cause a change in the value of connectivity index or average connectivity index, but the change will not necessarily be related to both values. This paper also defines the concepts of partial cubic connectivity enhancing node and partial cubic connectivity reducing node and some related results are proved. Furthermore, the concepts of cubic α −strong, cubic β − strong, cubic δ −weak edge, partial cubic α −strong and partial cubic δ −weak edges are utilized to identify areas most affected by a tsunami resulting from an earthquake. Finally, the research findings are compared with the existing methods to demonstrate their suitability and creativity.

Citation: Shi X, Kosari S, Hameed S, Shah AG, Ullah S (2024) Application of connectivity index of cubic fuzzy graphs for identification of danger zones of tsunami threat. PLoS ONE 19(1): e0297197. https://doi.org/10.1371/journal.pone.0297197

Editor: Yasuko Kawahata, Rikkyo University, JAPAN

Received: September 9, 2023; Accepted: January 1, 2024; Published: January 30, 2024

Copyright: © 2024 Shi et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper.

Funding: This work was supported by the National Key R & D Program of China (Grant 2019YFA0706402) and the National Natural Science Foundation of China under Grant 62172302, 62072129 and 61876047. There is no additional external funding received for this study.

Competing interests: The authors have declared that no competing interests exist.

1 Introduction

1.1 Motivation and contribution

Cubic fuzzy graphs have more advantageous representation as compared to interval-valued fuzzy graphs and fuzzy graphs because they depict the membership degree of vertices and edges in both interval and fuzzy number forms. This improved representation enables a deeper and more detailed comprehension of the connections and uncertainties present within the structure of the graph. The following features of strong and weak edges in CFG theory motivate us to present this paper:

In practical situations, some problems can be solved by using either FG or IVFG concepts, while more complex problems may require a combination of both. CFGs provide a useful tool to tackle such problems. For example, traffic flow modeling and earthquake modeling problems can be addressed with the help of CFGs.
Given the extensive applications of strong and weak edges in crisp and fuzzy graphs across various fields, it is worthwhile to investigate their relevance to CFGs as well.
It is observed that the definitions of cubic α −strong and cubic δ −weak edges for cubic graphs [ 42 ] are very strict. It may happen that a connected network may not have any such edges. In this situation, the decision making can be difficult in these connectivity problems. To overcome this problem, a more general model for strong edges has to be defined.

The study of strong, weak edges and connectivity index can be implemented in variety of decision-making problems.

Given the extensive importance and broad applications of cubic α −strong, cubic β −strong and cubic δ −weak edges within fuzzy networks, we have introduced the notion of partial cubic α −strong and partial cubic δ −weak edges for CFGs. These partial edges prove beneficial in addressing practical issues where the concept of cubic α −strong, cubic β −strong and cubic δ −weak edges may not be applicable. Specifically, these concepts come into play when the IVF − connectivity strictly exceeds or falls below the IVF −membership value of an edge, while the F − connectivity equates to the F −membership value of that edge and vice versa. In scenarios where we have information about the past, future and current conditions of a model or problem, we can represent the past condition as a lower interval-valued fuzzy membership, the future condition as an upper interval-valued fuzzy membership and the present condition as a fuzzy membership value. Our objective is to scrutinize the problem by deducing lower interval-valued fuzzy connectivity, upper interval-valued fuzzy connectivity and fuzzy connectivity. Furthermore, we aim to make new predictions based on this analysis. In these situations, the IVF − connectivity strictly exceeding or falling below the IVF −membership value of an edge occurs, while the F − connectivity aligns with the F −membership value of that edge and vice versa. To tackle this issue effectively, we can employ the concept of partial cubic α −strong and partial cubic δ −weak edges. Such problems frequently arise in the analysis of transportation networks, decision-making under uncertainty and optimization scenarios. Utilizing these partial cubic edges allows for a more accurate and detailed depiction of the connections between nodes or edges, enabling better modeling and evaluation of uncertain or imprecise relationships. It’s important to note that throughout this study, we specifically focused on simple connected CFGs. The primary contributions of this paper are outlined below.

Given the significant importance and numerous applications of strong and weak edges in fuzzy networks, the objective of this research paper is to investigate the concept of strong and weak edges in CFG.
To propose the concept of partial cubic α − strong and partial cubic δ − weak edges for CFG.
To study the connectivity index in CFG and to establish their bounds or exact expression for several families of CFG, e.g., for complete CFG, a CFG with underlying crisp tree and cubic fuzzy cycle.
To determine the effect on connectivity index of CFG on removal of an edge.
To define average connectivity index, partial cubic connectivity enhancing node (PCCEN) and partial cubic connectivity reducing node (PCCRN) for CFG.
To provide a more comprehensive understanding of the behavior of complex systems modeled by CFGs and to develop better strategies for addressing real-world problems such as earthquakes in certain areas by using cubic α − strong edges, cubic β −strong edges, cubic δ −weak edges, partial cubic α −strong edges and partial cubic δ −weak edges.
To demonstrate the novelty of our model, we compare our results with existing models.

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https://doi.org/10.1371/journal.pone.0297197.t001

2 Preliminaries

α −saturated if at each node of σ *, there are incident n ≥ 1 α − strong edges to it.
β -saturated if at each node of σ *, there are incident n ≥ 1 β strong edges to it.
Saturated if it is α − as well as β −saturated.
Unsaturated if it is neither α nor β saturated.

3 Partial cubic α − Strong and δ − Weak edges

The CF α − strong and CF δ − week edges are defined in [ 42 ]. But we note that there are CFGs which contain edges which are either IVF − α − strong and F − β − strong or IVF − β − strong and F − α − strong but not CF α − strong. These type of edges seem very close to CF α − strong edges and may be more useful in different CF connectivity problems. The following examples are helpful to understand this situation:

https://doi.org/10.1371/journal.pone.0297197.g001

It is clear that the edge fn is IVF − α − strong edge but F − β − strong edge. We can see that If we slightly increase the value of F -membership of edge fn , then it becomes CF α − strong edge. So we can say that it is very close to CF α − strong edge.

https://doi.org/10.1371/journal.pone.0297197.g002

It is clear that the edge nw is IVF − δ − weak and F − β − strong edge. Here if we slightly decrease the value of F -membership, then it becomes CF δ −weak edge. Above examples motivate to define the concept of partial cubic α − strong and partial cubic δ − weak edges.

Definition 8 For a CF edge t w −1 t w in CFG, if one of the following holds, then t w −1 t w is called partial cubic α − strong edge.

In Example 1, the edge fn satisfies condition 2 of above definition, so the edge fn is partial cubic α − strong edge.

Definition 9 For a CF edge t w −1 t w in CFG, if one of the following holds, then t w −1 t w is called partial cubic δ − weak edge.

In Example 2, the edge nw satisfies condition 2 of above definition, so the edge nw is a partial cubic δ − weak edge.

Partial α −saturated if at each node of σ *, there are incident n ≥ 1 partial α − strong edges to it.
Partial saturated if it is partial α −saturated as well as β −saturated.

4 Bounds for connectivity index of cubic fuzzy graph

ij is β -strong.
ij is partially δ - weak.

5 Average connectivity index of a cubic fuzzy graph

If (1) holds, then i is referred as IVF − connectivity reducing node, whereas if (2) is satisfied, then it is referred as F − connectivity reducing node. If both (1) and (2) are satisfied, then it is referred as connectivity reducing node.

If (1) holds, then i is referred as IVF − connectivity enhancing node, whereas if (2) is satisfied, then it is referred as F − connectivity enhancing node. If both (1) and (2) are satisfied, then it is referred as connectivity enhancing node.

6 Application to determine danger zone of tsunami threat

Natural disasters are events that are caused by natural phenomena and can have devastating consequences for the environment, human populations and infrastructure. They can take many different forms, including floods, hurricanes, earthquakes, tsunamis, tornadoes, wildfires and volcanic eruptions. One of the defining characteristics of natural disasters is their unpredictability. When natural disasters strike, they can cause widespread destruction and loss of life. They can also disrupt entire economies, causing significant financial losses and exacerbating social and political tensions.

Earthquakes are one of the most destructive and unpredictable natural disasters. An earthquake is a sudden rapid shaking of the ground caused by the movement of tectonic plates. It can cause significant damage to buildings and infrastructure, as well as trigger secondary hazards such as tsunamis, landslides and fires. The impact of earthquakes can widespread damage to buildings, roads, bridges and ports, as well as disruptions to essential services such as electricity and water. Secondary hazards such as tsunamis, landslides and fires can also exacerbate the impact of the disaster.

Earthquakes are a major natural hazard that can have a significant impact on communities and economies. To reduce the impact of earthquakes, it is important to invest in disaster risk reduction measures and emergency response planning, as well as to build infrastructure that is able to withstand earthquakes and other natural hazards.

Therefore, here we discuss the impact of earthquakes in certain areas by using cubic α −strong edges, cubic β −strong edges, cubic δ −weak edges, partial cubic α −strong edges and partial cubic δ −weak edges.

For this purpose, consider the problem in which an earthquake take place in deep ocean. A team from Pacific Tsunami Warning Center (PTWC) has to decide to find the region which is in danger zone of tsunami threat.

6.1 Tsunami threat model

With the help of CFG, a tsunami threat model is developed. In this tsunami threat model, vertices correspond to different areas with lower IVF -membership values indicating past tsunami threat values, upper IVF -membership values indicating future tsunami threat values and F -membership values indicating current tsunami threat values. The edges in this system represent the possibility of a danger zone arising due to a tsunami threat. By analyzing the strength of the connectedness between different areas, we can classify the types of danger zones into five categories: cubic α −strong zone, cubic β −strong zone, cubic δ −weak zone, partial cubic α −strong zone and partial cubic δ −weak zone. A cubic α −strong zone represents area with no tsunami threat, a partial cubic α −strong zone represents area with a very low tsunami threat, a cubic β −strong zone represents area with a low tsunami threat, a partial cubic δ −weak zone represents areas with a high tsunami threat and a cubic δ −weak zone represents area with a very high tsunami threat.

https://doi.org/10.1371/journal.pone.0297197.g003

https://doi.org/10.1371/journal.pone.0297197.t002

https://doi.org/10.1371/journal.pone.0297197.t003

https://doi.org/10.1371/journal.pone.0297197.t004

https://doi.org/10.1371/journal.pone.0297197.t005

It is noted that cubic α −strong zones are ( a 3 , a 5 ), ( a 3 , a 4 ), ( a 4 , a 6 ), cubic β −strong zones are ( a 1 , a 2 ), ( a 1 , a 3 ), cubic δ −weak zones are ( a 4 , a 5 ), ( a 5 , a 6 ), there is only one partial cubic α −strong zone which is ( a 2 , a 4 ) and partial cubic δ − weak zone is ( a 2 , a 3 ) in CFG system. The classification of areas with tsunami threat into different zones, will be helpful to interpret the situation of tsunami threat in areas due to earthquake. Based on the categorization of different zones according to the tsunami threat level the level of planning requires variation. In the cubic α −strong zone with no tsunami threat minimal planning is needed focusing on general disaster preparedness measures. A partial cubic α −strong zone requires moderate planning including early warning systems and resilient infrastructure. A cubic β −strong zone demands a higher level of planning with comprehensive emergency response plans and coastal protection measures. In a partial cubic δ −weak zone, extensive planning is necessary involving drills, evacuation centers and strict building codes. A cubic δ −weak zone representing a very high tsunami threat, requires the utmost level of planning including tsunami-resistant structures and advanced warning systems. Overall, planning efforts must align with the level of tsunami threat in each zone to ensure effective disaster risk reduction and mitigation measures. It is important to note that throughout this study, we specifically focused on simple connected CFGs. The concept of partial cubic α −strong and δ −weak edges is more advantageous compared to cubic strong and weak edges. This is because sometimes we encounter a problem or graph structure where the IVF -connectivity is either strictly less or greater than the IVF -membership value of an edge, while the F -connectivity equals to the F -membership value of that edge and vice versa. In such situations, the concept of cubic strong and weak edges fails to provide us with any relevant information about the nature of that edge, leading to difficulty in understanding it. In these conditions, the concept of partial cubic α −strong and δ −weak edges plays an important role by providing us with information about the nature of that edge. Hence, the concept of partial cubic α −strong and δ −weak edges is more beneficial compared to cubic strong and weak edges.

7 Comparative analysis

8 Conclusion

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Introduction

Preliminaries

Definition 1 7

Definition 2 60

Definition 3 48

Definition 4 51

Definition 5 51

t-intuitionistic fuzzy graph

Definition 6

Definition 7

Definition 8

Definition 9

Proposition 1

Definition 10

Proposition 2.

Corollary 1.

Corollary 2.

Operations on t-intuitionistic fuzzy graph

Definition 11.

Definition 12.

Proposition 3.

Definition 13.

Definition 14.

Proposition 4.

Definition 15.

Definition 16.

Proposition 5.

Definition 17.

Example 10.

Definition 18.

Proposition 6.

Isomorphism of t-intuitionistic fuzzy graphs

Definition 19.

Definition 20.

Definition 21.

Definition 22.

Example 11.

Complement of t-intuitionistic fuzzy graph

Definition 23.

Example 12.

Definition 24.

Proposition 7.

Proposition 8.

Proposition 9.

Proposition 10.

Proposition 11.

Proposition 12.

Proposition 13.

Application of t-intuitionistic fuzzy graph

Comparative analysis

Data availability

Acknowledgements

Author information

Contributions

Corresponding authors

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