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In this paper, we established the concept of chordless path in a fuzzy graph, fuzzy strong chordal graph and studied some of its properties. Also we proved that the center of a connected fuzzy strong chordal graph is connected and it is again a fuzzy strong chordal graph.

In this paper we discuss several operations on fuzzy graphs such as union, join, composition, Cartesian product and study their domination parameters.

Let S be a semigroup. This paper studies the intersection graphs of fuzzy semigroups. It is shown that the fuzzy intersection graph Int(G(S)), of S, is complete if and only if S is power joined. If Γ(S) denotes the set of all fuzzy right ideals of S, then the fuzzy intersection graph Int(Γ(S)) is complete if and only if S is fuzzy right uniform. Moreover, it is shown that Int(Γ(S)) is chordal if and only if for a,b,c,d ∈ S, some pair from {a,b,c,d} has a right common multiple property. It is also shown that if Int(G(S)) is complete and S has the acc on subsemigroups, then S is cyclic.

Let $G=(\sigma ,\mu )$ be a fuzzy graph on a finite set $V.$ Let $r,s\in [0,1]$ and $r<s.$ A fuzzy subset ${\sigma }_1$ of $\sigma$ is called an $(r,s)$-fuzzy dominating set ($(r,s)$-FD set) of G if

Directed fuzzy networks are introduced in this paper. They are normalized node capacitated networks and provide a good platform to model different types of complicated flows in nature. A directed fuzzy network version of Menger’s theorem and the celebrated Max flow Min cut theorem are also provided. Since the maximum flow through minimum number of directed internally disjoint paths is important in quality of service (QoS) problems in networking, the results in this paper can be applied to a wide variety of problems.

A vague graph is a generalized structure of a fuzzy graph that gives more precision, exibility, and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, the concepts of eccentricity of nodes, radius and diameter of vague graphs are introduced. The special types of graphs such as eccentrice and antipodal vague graphs are investigated. Then, the relation between eccentrice and antipodal vague graphs are discussed. Finally, an application of eccentrice and antipodal vague graphs in human traffickingn studied.

Edges of a fuzzy graph are mainly classified into $\alpha$, $\beta$ and $\delta$. In this paper, we study certain saturation counts with respect to the classification of edges. Characterizations for fuzzy cycles, fuzzy trees, blocks in fuzzy graphs and complete fuzzy graphs are also obtained using saturation counts.

Connectivity is the most important aspect of a dynamic network. It has been widely studied and applied in different perspectives in the past. In this paper, constructions of $t$-connected fuzzy graphs for an arbitrary real number $t$ and average fuzzy vertex connectivity of fuzzy graphs are discussed. Average fuzzy vertex connectivity of fuzzy trees, fuzzy cycles and complete fuzzy graphs are studied. The concept of a uniformly $t$-connected fuzzy graph is introduced and characterized towards the end. An application related to human trafficking is also discussed.

It is much more practical to use the $t$-norm than the minimum in defining the fuzzy graph. Hence, the concept of $t$-fuzzy graph was introduced by Mordeson et al. in [J. N. Mordeson and S. Mathew, $t$-norm Fuzzy Graphs, New Mathematics and Natural Computation14(1) (2018) 129–143]. In this paper, we concentrate on the relatively new concept of fuzzy adjacency matrix of that $t$-fuzzy graphs. The purpose of this paper is to further develop a product of this fuzzy adjacency matrix based on $t$-fuzzy graphs. Moreover, two kinds of Boolean matrix are defined and relation between Normal Boolean matrix, product of adjacency matrix and 1-level cut Boolean matrix is obtained.

In this paper, we investigate the fuzzification of zero forcing process. For this, first we introduce a new embedding of a graph $G$ by considering a minimal zero forcing set of $G$ and an arbitrary list of maximal forcing chains of this zero forcing set. Then we get a comparison between zero forcing sets of a graph by using fuzzy concepts. Finally, we give an application for this procedure.

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.

Fuzzy graph theory is finding an increasing number of application in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Special fuzzy graph can be obtained from two given fuzzy graphs using the operations beta products. In this paper, we introduce the notions of some kinds of beta product of two fuzzy graphs. The concept of strong, regular and complement of ${\beta }_i$-product of two fuzzy graphs and relation between them are also obtained. At the end, an application with a cryptographic object is said to be using the ${\beta }_i$-product of fuzzy graphs.

Spanness of fuzzy graph is introduced. By spanness, a new vulnerability parameter, span integrity is defined in fuzzy graph. The span integrity values are found for path, cycle, complete fuzzy graph, complete bipartite fuzzy graphs. Path and cycle with node strength sequence are discussed. Brain network is modeled as a fuzzy graph and Span integrity is applied to the brain network. Span integrity of fuzzy brain network is calculated for before and after meditation models. The results are compared and the improvement in the stability of the brain network is shown.

Containers in fuzzy graphs are studied in this paper. A container in a graph is a family of internally disjoint paths between a pair of vertices. Equivalently, a container in a fuzzy graph is a collection of internally disjoint strongest paths. Strongest paths often provide maximum contribution to a network’s traffic flow. Therefore analysis of strongest paths and containers helps examining the dynamics and complexities of networks. Apart from certain structural properties of containers, two classic results on graph degrees are also generalized to fuzzy setup. Spanning containers of blocks, fuzzy trees and locamin cycles are discussed. An algorithm to identify the maximum and minimum capacity of containers and an application related to human trafficking are also proposed.

Fuzzy graph theory has an important role in modeling uncertainty in the real environment. In this work, a new approach is proposed concerning the graph coloring problem of a fuzzy graph using Gröbner basis, and a generalization of the classical crisp chromatic number of a graph. This approach is established on the successive coloring functions of the crisp graphs. Using this method, we can determine whether or not, for a fixed $k$, the given graph can be colored. If such a coloring exists, the number of distinct $k$-colorings and chromatic number in the given fuzzy graph can be obtained. As an application, we analyze the traffic lights problem by using this method.

In this paper, based on the outcomes from the published literature, a brief survey of domination of fuzzy graphs is presented. Discovering what has been accomplished for a particular kind of domination and staying aware of new revelations are difficult due to the huge number of papers and on the grounds that a significant number of the papers have been published in journals that are not broadly accessible. This survey is based on the papers that I could find on the domination of fuzzy graphs. For the comfort of the reader, the overview incorporates some basic definitions and techniques used to study domination of fuzzy graphs.

Graph theory has various applications in computer science, such as image segmentation, clustering, data mining, image capturing, and networking. Fuzzy graph (FG) theory has been widely adopted to handle uncertainty in graph-related problems. Interval-valued picture fuzzy graphs (IVPFGs) are a generalization of FGs, interval-valued FGs, intuitionistic fuzzy graphs (IFGs), and interval-valued IFGs. This paper introduces the concept of interval-valued picture fuzzy sets to graph theory and presents a new type of graph called the IVPFG. Within this framework, we define the degree, order, and size of IVPFGs. The paper further explores various operations on IVPFGs, including the Cartesian product, composition, join, and union. The paper delves into the properties of these operations, providing proofs and examples to support the findings. By studying the operations on IVPFGs, we can gain insights into their behavior and leverage this knowledge for solving graph-based problems in the presence of uncertainty. Also, an application regarding merging of community is provided.

In this paper the questions of definition optimum allocation of the service centers of some territory are observed. It is supposed that territory is described by fuzzy graph. In this case a task of definition optimum allocation of the service centers may be transformed into the task of definition of fuzzy antibases of fuzzy graph. The method of definition of fuzzy antibases is considered in this paper. The example of founding optimum allocation of the service centers as definition of fuzzy antibases is considered too.