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In this research study, we first define the strong degree of a vertex in an $ m $-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete $ m $-polar fuzzy graphs. Further, we introduce the concept of $ m $-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete $ m $-polar fuzzy graphs. We discuss connectivity parameters in $ m $-polar fuzzy graphs with precise examples, and we investigate the $ m $-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an $ m $-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as $ \epsilon_A $-reachable vertices in $ m $-polar fuzzy graphs, $ \epsilon_A $-connected $ m $-polar fuzzy graphs, or $ \epsilon_A $-connected $ m $-polar fuzzy subgraphs (in case the $ m $-polar fuzzy graph itself is not $ \epsilon_A $-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.

Many papers have been published for finding eigenvalues and eigenvectors of a fuzzy matrix (FM) (i.e. matrix with membership values) as well a fuzzy graph. But, all papers determine such parameters based on conventional arithmetic operators, though the valid operations on FMs are max–min. To the best of our knowledge no papers are published to find eigenvalues and eigenvectors based on max–min operators. In this paper, a novel technique is adapted to find the eigenvalues and eigenvectors of an interval-valued fuzzy graph (IVFG) using max–min operators. The energy of an IVFG is defined and computed using max–min operators. Finally, an application of eigenvalues of an IVFG is discussed for the ecological system. In ecology, the amount of food consumed by a predator from the preys is represented as an interval valued fuzzy membership values which is natural as the consumption of food for a predator from preys is uncertain. So this application is very much appropriate for eigenvalues and eigenvectors as well as energy of an IVFG.

Connectivity can be used to measure the strength and combined power of a connected network system. Randic index of graph is one such parameter and it can measure the total combined power of a connected graphical transmission system. For two opposite sided opinion of vertices as well as edges in a bipolar fuzzy graph, it can measure the uncertainty of vertices and edges along positive and negative sides. In this article, the Randic index of bipolar fuzzy graph and bipolar fuzzy subgraph are introduced with their properties. The upper and lower boundaries of Randic index of bipolar fuzzy graphs are studied with some isomorphic properties. Randic index of directed bipolar fuzzy graphs are introduced. Several formula’s are presented to calculate the Randic index of different types of regular bipolar fuzzy graphs and bipolar fuzzy cycles. Finally, two real life applications of Randic index in bipolar fuzzy graphs are described.

The main objective of this research article is to classify different types of m-polar fuzzy edges in an m-polar fuzzy graph by using the strength of connectedness between pairs of vertices. The identification of types of m-polar fuzzy edges, including α-strong m-polar fuzzy edges, β-strong m-polar fuzzy edges and δ-weak m-polar fuzzy edges proved to be very useful to completely determine the basic structure of m-polar fuzzy graph. We analyze types of m-polar fuzzy edges in strongest m-polar fuzzy path and m-polar fuzzy cycle. Further, we define various terms, including m-polar fuzzy cut-vertex, m-polar fuzzy bridge, strength reducing set of vertices and strength reducing set of edges. We highlight the difference between edge disjoint m-polar fuzzy path and internally disjoint m-polar fuzzy path from one vertex to another vertex in an m-polar fuzzy graph. We define strong size of an m-polar fuzzy graph. We then present the most celebrated result due to Karl Menger for m-polar fuzzy graphs and illustrate the vertex version of Menger’s theorem to find out the strongest m-polar fuzzy paths between affected and non-affected cities of a country due to an earthquake. Moreover, we discuss an application of types of m-polar fuzzy edges to determine traffic-accidental zones in a road network. Finally, a comparative analysis of our research work with existing techniques is presented to prove its applicability and effectiveness.

The perception of Fuzzy Proper Bi-Magic labeling for Tadpole Graph also-ran newly derived and the properties of fuzzy bi-magic of Tadpole graph has been discussed. The property of Fuzzy Bi-Magic of Tadpole graph is the sum of the labels on the vertices and the labels of their incident edges is one of the constants either k1 or k2 but k2 is unique

In a fuzzy graph , for a subset of , the representation of with respect to in terms of strength of connectedness of vertices are distinct then is called the fuzzy resolving set of . In this article, we discuss the properties of fuzzy resolving set and fuzzy resolving number. And also proved some theorems and properties of fuzzy resolving number in fuzzy cyclic and fuzzy labeling graphs.

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.

Fuzzy graph became a nice tool for modeling and designing a network that contains indeterminacy. Public transportation has become a necessity in Bantul Regency because it is one of the regions that...

Fuzzy graphs are an individual of application tools in the area of mathematics, which permit the users to define the relative between concepts because the wildlife of fuzziness is satisfactory for any situation. They are helpful to give more exactness and suppleness to the classification as associated with the traditional models. A topological structure is a set model for graphs. The main purpose of this paper is to introduce a new kind of fuzzy topological structures in terms of fuzzy graphs called fuzzy topological graphs due to a class of fuzzy subsets, and some of their properties are investigated. Also, a new procedure to calculate the number of edges in fuzzy graphs will be defined. Further, we consider the concept of a homeomorphic between fuzzy topological graphs as a fuzzy topological property that can be used to prove the isomorphic between fuzzy graphs. Moreover, an algorithm based on the proposed operations that build some fuzzy topological graphs will be presented. Finally, we give a new method to explain the homeomorphic between some fuzzy topological graphs which will be applied in smart cities.

Due to the existence of two opposite sided opinions in bipolar fuzzy graphs, the positive and negative communication between all pair of vertices always may not be strong. So, the connectivity sustain is one of the major important part in a bipolar fuzzy network system. First, Wiener index for a bipolar fuzzy graph is introduced and explained their properties. Second, the terms Wiener absolute index is created based on the total accurate connectivity between all the pair of vertices and in the whole bipolar fuzzy graph. Third, the behavior of Wiener absolute index is visualized in several bipolar fuzzy graphs like bipolar fuzzy forest, bridge, and tree. Fourth, a comparative discussion between connectivity index and Wiener index in bipolar fuzzy graphs are established. Finally, an application of all these thought is displayed in regular journeys from the city Paris to Brest.

Up to now, there have been a lot of research results about multi-attribute decision making problems by fuzzy graph theory. However, there are few investigations about multi-attribute decision making problems under the background of indecisiveness. The main reason is that the difference of cognition and the complexity of thinking by decision makers, for the same question have different opinions. In this paper, we proposed a hesitant fuzzy hypergraph model based on hesitant fuzzy sets and fuzzy hypergraphs. At the same time, some basic graph operations of hesitant fuzzy hypergraphs are investigated and several equivalence relationship between hesitant fuzzy hypergraphs, hesitant fuzzy formal concept analysis and hesitant fuzzy information systems are discussed. Since granular computing can deal with multi-attribute decision-making problems well, we considered the hesitant fuzzy hypergraph model of granular computing, and established an algorithm of multi-attribute decision-making problem based on hesitant fuzzy hypergraph model. Finally an example is given to illustrate the effectiveness of the algorithm.

This paper comprises the introduction of weighted mean products of fuzzy graph structures (FGSs) to construct weighted mean fuzzy graph structures (WMFGSs) with the help of optimization parameter, and establish some novel results after validating with examples, accordingly. The notions of regular and $m\mu _{k}$ -regular FGSs are described, where $m\in$ ]0, 1] represents the degree of all vertices in $\widetilde {G}$ under mapping $\mu _{k}$ , and develop certain properties of regular WMFGSs. In addition, we create a flowchart to present common application procedures of fuzzy graphical frameworks to classify the advanced city out of some important Pakistani cities subject to certain parameters.

The present study aims to introduce new methods to achieve optimal matching in fuzzy graphs and classify the fuzzy sizes of the edges and vertices of a matching called “matching number”. Matching numbers in a fuzzy graph are not only a direct tool in improving existing matching optimization algorithms, but also can be used to build optimization algorithms based on the vertices of a matching. Thus, introduce the properties of matching numbers which are useful in constructing and solving edge-fuzzy and vertex-fuzzy maximization problems.

In this paper, the notion of picture fuzzy competition graph along with its few generalizations such as m-step picture fuzzy competition graphs, picture fuzzy economic competition graphs and picture fuzzy competition hypergraphs are introduced. Some related picture fuzzy graphs including picture fuzzy m-step neighborhood graph, picture fuzzy m-step economic competition graph and picture fuzzy k-competition hypergraphs are introduced. Some properties of these graphs have been investigated. Finally, applications of m-step picture fuzzy competition graphs and picture fuzzy competition hypergraphs are presented in several fields such as in education system, ecosystem, business market and job competition.

In this paper, we obtain a reversible regular hypergroup from fuzzy sets by using a t-norm. Some properties of isomorphism of t-fuzzy graphs are considered and we show that a t-fuzzy subgroup can be associated with a t-fuzzy graph. Finally, using the group of automorphisms of fuzzy graph, we explain the relationship between the hypergroup and the t-fuzzy subgroup with the t-fuzzy graph.

Rosenfeld [A. Rosenfeld, Fuzzy Graphs, Fuzzy Sets and Their Applications, eds. L. A. Zadeh, K. S. Fu and M. Shimura (Academic Press, New York, 1975), pp. 77–95.] defined the fuzzy relations on the fuzzy sets and developed the structure of fuzzy graph, as a graph with a membership degree (between zero and one) for the vertices and edges such that the membership degree of every edge is less than or equal to the minimum of the membership degree of its endpoints. Although this model of graph has many applications in the real life, it fails to solve a lot of problems, which we can use graph for its representation. This paper aimed to demonstrate a new type of graph with a membership degree (between zero and one) for the vertices and edges so that the membership degree of every edge becomes more than or equals the minimum of the membership degrees of its endpoints. This new type of graph is called inverse fuzzy graph “or” I-fuzzy graph, which can play a role in solving many problems which are not solved by fuzzy graph.

Shortest path problem (SPP) is a fundamental and well-known combinatorial optimization problem in the area of graph theory. In real-life scenarios, the arc weighs in a shortest path of a network/graph have the several parameters which are very hard to define exactly (i.e., capacity, cost, demand, traffic frequency, time, etc.). We can incorporate the fuzziness into a graph to handle this type of uncertain situation. In this manuscript, we propose the idea of constrained SPP (CSPP) in fuzzy environment. CSPP has an useful real-life application in online cab booking system. The main motivation of this study is to determine a path with minimal cost where traveling time within two locations does not more than predetermined time. We can not predicate the exact time and cost of the path due to uncertain traffic scenarios and another unexpected reasons; still, the geometrical distance between the locations is fixed. Here, we use trapezoidal fuzzy number to describe the edge weight of a fuzzy network/graph for CSPP. We define this CSPP as fuzzy CSPP (FCSPP). The utility of FCSPP is described in several real-life scenarios. We propose a mathematical formulation for the FCSPP and an algorithm is proposed for solving the FCSPP. We describe an application of our proposed algorithm on an online cab booking system.

In this paper, we propose some developments in fuzzy graph theory. An original notion of a T-spherical fuzzy graph is presented as a commonality of fuzzy graph, an intuitionistic fuzzy graph and a picture fuzzy graph. The originality, the imperativeness and the importance of this notion are discussed by showing some results, giving examples and graphical analysis. Some theoretical terms of graphs such as a T-spherical fuzzy subgraph, a complement of T-SFG, degree of T-SFG are clarified and their attributes and aspects are analyzed. Interestingly, three types of decision-making problems using the framework of T-SFGs are studied. These problems include the problem of the shortest path, the safe root for an airline journey and the problem of supply chain management in a T-spherical fuzzy network. The comparison of this new approach towards these problems with existing approaches is also established. A new algorithm is put forward in the event of T-SFGs and is used to seek out the shortest path problem. The overall analysis of the suggested notion under the prevailing theory is conducted. The advantages of the proposed approach are discussed based on the existing tools and a short comparison of the new with existing tools is established.

Highlights •The notion of DHFGs is put forward via the definition of DHFGs and strong DHFGs.•Some common operational laws and mapping relationships of DHFGs are explored.•A two-stage DHF MADM algorithm by virtue of DHFGs is designed for addressing DHF MADM problems.•The applicability of the DHF MADM algorithm is verified by a case study and a comparative analysis.In real world, there exist a kind of Multi-Attribute Decision Making (MADM) problems in which correlations and prioritization relationships of attributes can be found. The notion of Fuzzy Graphs (FGs) performs well when expressing correlations between attributes via edges between vertices in fuzzy information systems, which makes it possible for addressing correlational MADM problems. In addition, as a significant information depiction tool, Dual Hesitant Fuzzy Sets (DHFSs) generalize traditional Intuitionistic Fuzzy Sets (IFSs) to the context of hesitations, thus DHFSs excel in recording numerous imprecise and hesitant information. In this paper, we aim to investigate FGs in the Dual Hesitant Fuzzy (DHF) background and further explore efficient approaches to complicated MADM situations. Following the above motivation, we first develop the definition of Dual Hesitant Fuzzy Graphs (DHFGs). Then, some common operational laws and mapping relationships are put forward. Afterwards, we construct a two-stage MADM approach by means of DHFGs for addressing complicated MADM situations with correlations and prioritization relationships. At last, a realistic case study along with a comparative analysis is presented to show the applicability of the established methodology. The proposed model and approach are conducive to addressing DHF MADM problems that are characterized by correlations and prioritization relationships between attributes at the same time, and further enrich the family of DHF MADM methods to a large extent.

In this article, a new idea of fuzzy fractional coloring of fuzzy graph is presented and fuzzy fractional chromatic number is defined. A relationship between fuzzy fractional chromatic number and fuzzy fractional clique number is established. Some properties of fuzzy chromatic number of fuzzy graphs and fuzzy fractional chromatic number of fuzzy graphs are proved and the concept of k-strong adjacent vertices is introduced. Fuzzy chromatic number and fuzzy fractional chromatic number have been calculated on lexicographic product of two fuzzy graphs. Also, fuzzy chromatic number, independence number and fuzzy fractional chromatic number have been investigated on disjoint union of two fuzzy graphs. Lastly, a real life application of fuzzy fractional coloring on fuzzy graph is discussed.

We focus on fuzzy graphs with crisp vertex and fuzzy edge sets. A concept of the fuzzy chromatic number of these graphs based on fuzzy independent vertex set is used in this paper. A modified algorithm called a fuzzy chromatic algorithm is developed to find the fuzzy chromatic number of union of fuzzy graphs. Running time and complexity of the algorithm are also analyzed. Furthermore, we investigate some properties of the fuzzy chromatic number of union of fuzzy graphs. Finally, an application of the fuzzy chromatic number to determine the number of phases of an integrated traffic light system is proposed. We get different phases with different degrees of safety.

Connectivity parameters have important role in the study of networks in the physical world. Wiener index is one such parameter with several applications in chemistry and network theory. In this article Wiener index of various fuzzy graph structures like fuzzy trees and fuzzy cycles are discussed. Relationship between connectivity index and Wiener index of a fuzzy graph is also studied. For the effective computation of Wiener Index of a fuzzy graph an algorithm is proposed. A real life application related with human trafficking and illegal immigration is presented towards the end.

An autocatalytic set (ACS) is a graph. On the other hand, the Potential Method (PM) is an established graph based concept for optimization purpose. Firstly, a restricted form of ACS, namely, weak autocatalytic set (WACS), a derivation of transitive tournament, is introduced in this study. Then, a new mathematical concept, namely, fuzzy weak autocatalytic set (FWACS), is defined and its relations to transitive PM are established. Some theorems are proven to highlight their relations. Finally, this paper concludes that any preference graph is a fuzzy graph Type 5.

New notion of dimension as set, as two optimal numbers including metric number, dimension number and as optimal set are introduced in individual framework and in formation of family. Behaviors of twin and antipodal are explored in fuzzy(neutrosophic) graphs. Fuzzy(neutrosophic) graphs, under conditions, fixed-edges, fixed-vertex and strong fixed-vertex are studied. Some classes as path, cycle, complete, strong, t-partite, bipartite, star and wheel in the formation of individual case and in the case, they form a family are studied in the term of dimension. Fuzzification(neutrosofication) of twin vertices but using crisp concept of antipodal vertices are another approaches of this study. Thus defining two notions concerning vertices which one of them is fuzzy(neutrosophic) titled twin and another is crisp titled antipodal to study the behaviors of cycles which are partitioned into even and odd, are concluded. Classes of cycles according to antipodal vertices are divided into two classes as even and odd. Parity of the number of edges in cycle causes to have two subsections under the section is entitled to antipodal vertices. In this study, the term dimension is introduced on fuzzy(neutrosophic) graphs. The locations of objects by a set of some junctions which have distinct distance from any couple of objects out of the set, are determined. Thus it’s possible to have the locations of objects outside of this set by assigning partial number to any objects. The classes of these specific graphs are chosen to obtain some results based on dimension. The types of crisp notions and fuzzy(neutrosophic) notions are used to make sense about the material of this study and the outline of this study uses some new notions which are crisp and fuzzy(neutrosophic).

The prime objective of this paper is to introduce a new model of threshold graphs in the complex fuzzy environment that has the capability of handling two-dimensional vague data. The proposed model is called as complex fuzzy threshold graph (CFTG). We introduce certain concepts related to CFTGs including, complex fuzzy alternating 4-cycle, complex fuzzy threshold dimension and complex fuzzy partition number of complex fuzzy graphs. We illustrate these concepts with examples. Further, we investigate some of their properties and discuss proficiency of the presented approach by investigating potential applications related to water source controlling system and internet service providers.

A new domination parameter in a fuzzy digraph is proposed to espouse a contribution in the domain of domination in a fuzzy graph and a directed graph. Let GD*=V,A be a directed simple graph, where V is a finite nonempty set and A=x,y:x,y∈V,x≠y. A fuzzy digraph GD=σD,μD is a pair of two functions σD:V→0,1 and μD:A→0,1, such that μDx,y≤σDx∧σDy, where x,y∈V. An edge μDx,y of a fuzzy digraph is called an effective edge if μDx,y=σDx∧σDy. Let x,y∈V. The vertex σDx dominates σDy in GD if μDx,y is an effective edge. Let S⊆V, u∈V\S, and v∈S. A subset σDS⊆σD is a dominating set of GD if, for every σDu∈σD\σDS, there exists σDv∈σDS, such that σDv dominates σDu. The minimum dominating set of a fuzzy digraph GD is called the domination number of a fuzzy digraph and is denoted by γGD. In this paper, the concept of domination in a fuzzy digraph is introduced, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled.

We develop indices that can be used to measure the susceptibility of a route to human trafficking. The susceptibility is based on data providing a measure of a country’s vulnerability for and its government response to human trafficking. The data was normalized and the Pearson correlation coefficient was used to determine the correlation between five types of government response and five types of vulnerability. It was determined that comparing government response values with vulnerability values yielded a negative correlation. Four routes through the Americas to the United States were examined. The indices of two of the measures agree on all four routes.

Pythagorean fuzzy graph, a broadly used extension of fuzzy and intuitionistic fuzzy graph, is helpful in representing structural relationships between several objects where the relation between these objects is vague, while the Dombi operators with operational parameters have excellent flexibility. Utilizing these two concepts, this research paper proposes the novel concept of Pythagorean Dombi fuzzy graphs (PDFGs). Basically, graph terminology is employed for introducing Pythagorean fuzzy analogs of various fundamental graphical ideas using Dombi operator. Further, under Pythagorean Dombi fuzzy environment, regular, totally regular, strongly regular and biregular graphs are defined with appropriate illustration and some of their crucial properties are examined. Meanwhile, the notion of edge regularity of PDFG is also initiated with substantial characteristics. Finally, a numerical example related to evaluation of appropriate ETL software for a business intelligence project is presented to better understand PDFGs.

The median problem on a graph is an important field of facility location problem in which the aim is to find the best place among vertices and interior points of edges such that the summation of weighted distances are minimized. This point is called absolute median point. Beforehand, it was shown that the absolute median of a graph is always at a vertex of it. This theorem is called vertex optimality theorem. In this paper, a graph with fuzzy weights is considered. Then, the fuzzy vertex optimality theorem is proved for this problem. That is, it is proved that an absolute median of this fuzzy graph is always at a vertex of it. Thereafter, this theorem is proved for the general form problem where the weights of vertices and the lengths of the arcs are all fuzzy numbers. It is worthwhile to note that this is the first time that this theorem is proved for fuzzy location problems. Also, by proving this fuzzy theorem and applying it on existing algorithms for obtaining median point, the computational complexity is reduced. Finally, by solving some numerical examples, the concept of these proved theorems and their proofs are illustrated.