Types of arcs in a fuzzy graph

In this chapter, the authors introduce some basic definitions related to fuzzy graphs like directed and undirected fuzzy graph, walk, path and circuit of a fuzzy graph, complete and strong fuzzy graph, bipartite fuzzy graph, degree of a vertex in fuzzy graphs, fuzzy subgraph, etc. These concepts are illustrated with some examples. The recently developed concepts like fuzzy planar graphs are discussed where the crossing of two edges are considered. Finally, the concepts of fuzzy threshold graphs and fuzzy competitions graphs are also given as a generalization of threshold and competition graphs.

Strong arcs in fuzzy graphs

The distance degree sequence and distance neighborhood degree sequence are two sequences in fuzzy graphs. In graph theory, the distance degree sequence of a vertex involves the concept of the number of vertices at various distances from the given vertex of a graph. The concept of distance degree sequence of a vertex in fuzzy graph is developed by taking into account the membership values of the edges which are incident on the vertices at various distances from the given vertex of fuzzy graphs, and the distance neighborhood degree sequence considers the membership values of the vertices at various distances. The drastic sum of fuzzy graphs is an operation in fuzzy graph theory, which is the extension of the standard operations on graphs using the drastic sum of fuzzy sets. The drastic sum provides a way to combine two fuzzy graphs into a new fuzzy graph by using a specific rule for combining the membership values of edges and vertices between two fuzzy graphs. In this paper, the distance degree sequence and distance neighborhood degree sequence of the vertices in the drastic sum of two fuzzy graphs on star graphs are obtained in terms of the parameters of the given graphs by considering all possible cases of drastic sum in detail.

A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

Let (V, μ, ρ) be a fuzzy graph. We now provide two popular ways of defining the distance between a pair of vertices. One way is to define the “distance” dis(x,y) between x and y as the length of the shortest strongest path between them. This “distance” is symmetric and is such that dis(x,x) = 0 since by our definition of a fuzzy graph, no path from x to x can have strength greater than μ(x), which is the strength of the path of length 0. However, it does not satisfy the triangle property, as we see from the following example. Let V = {u, v, x, y,z}, ρ(x, u) = ρ(u, v) = ρ(v, z) = 1 and ρ(x, y) = ρ(y, z) = 0.5. Here any path from x to y or from y to z has strength ≤ 1/2 since it must involve either edge (x,y) or edge (y, z). Thus the shortest strongest paths between them have length 1. On the other hand, there is a path from x to z, through u and v, that has length 3 and strength 1. Thus dis(x,z) = 3 > 1 + 1 = dis(x,y) + dis(y, z) in this case.KeywordsHamiltonian PathFuzzy SubsetFuzzy RelationConnected SubgraphFuzzy GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

AN INTRO TO $\delta_d$-FUZZY GRAPHS

In graph theory, the Randic index ( R ) is a topological graph invariant widely used as a physicochemical descriptor in the mathematical modeling of molecular structures. However, traditional molecular graphs fail to capture the heterogeneity of chemical bonds, since they treat all edges as uniform, ignoring variations in bond lengths and strengths. To overcome this limitation, we adopt a fuzzy graph structure, which provides a more realistic framework for representing molecular interactions. This dual perspective developing rigorous theoretical bounds for the Randic index on fuzzy graph classes and simultaneously applying them to real pharmaceutical compounds ensure both mathematical novelty and practical relevance. We derive analytical bounds for the Randic index over fuzzy versions of complete graphs, paths, stars, and cycles, as well as for graph operations such as Cartesian product, union, join, and composition, confirming the sharpness of the results in each case. To demonstrate the applicability, we compute the Randic index for fuzzy graph representations of Parkinson’s disease drugs (Levodopa, Procyclidine, Trihexyphenidyl, and Apomorphine). The findings indicate that the Randic index, within the fuzzy graph framework, reliably estimates key physicochemical properties such as polarizability, molar refractivity, surface tension, and molar volume, highlighting the strength of combining theoretical results with drug modeling applications.

Corona product of two fuzzy graphs which is analogous to the concept corona product operation in crisp graph theory is defined. The degree of an edge in corona product of fuzzy graphs is obtained. Also, the degree of an edge in fuzzy graph formed by this operation in terms of the degree of edges in the given fuzzy graphs in some particular cases is found. Moreover, it is proved that corona product of two fuzzy graphs is effective when two fuzzy graphs is effective fuzzy graphs.

Generally, we could efficiently analyze the inexact information and investigate the fuzzy relation by applying the fuzzy graph theory. We would extend the fuzzy graph theory, and propose a fuzzy node fuzzy graph. Since a fuzzy node fuzzy graph is complicated to analyze, we would transform it to a simple fuzzy graph by using T-norm family. In addition, to investigate the relations between nodes, we would define the fuzzy contingency table. In this paper, we would discuss about five subjects, (1) new T-norm Uesu product, (2) fuzzy node fuzzy graph, (3) fuzzy contingency table, (4) entropy measures of fuzziness and (5) decision analysis of the optimal fuzzy graph GÉ0 in the fuzzy graph sequence {GÉ}. By using the fuzzy node fuzzy graph theory, the new T-norm and the fuzzy contingency table, we could clarify the relational structure of fuzzy information. According to the decision method in section 2, we could find the optimal fuzzy graph GÉ0 in the fuzzy graph sequence {GÉ}, and clarify the structural feature of the fuzzy node fuzzy graph. Moreover, we would illustrate its practical effectiveness with the case study concerning sociometry analysis.

In this paper the idea of temporal reachability in fuzzy directed graph is introduced. Temporality in fuzzy graphs is a generalization of a fuzzy graph on temporal relations, where the incidence property of graph vertices is changed in the discrete time. Reachability in such temporal graph structures refers to the ability to get from one vertex to another within different time moments and the reachability relation on fuzzy directed graph is the transitive closure on the edge set. Efficient processing queries and graph structured information under uncertainty is an important research topic especially in XML databases, geographical information systems (GIS), web mining and etc. Fuzzy temporal graph models are usually applied in geographical information systems to formalize relations between objects and to check connections between them. A method for searching temporal reachability in fuzzy graph structure is proposed, an example of finding a fuzzy reachability set in temporal graph is considered as well.

In this manuscript, we introduce a few new types of dominations in intuitionistic fuzzy directed graphs (IFDGs) based on different types of strong arcs (SAs). Our work is not only a direct extension of domination in directed fuzzy graphs (DFGs) but also fills the gap that exists in the literature regarding the dominations in different extended forms of fuzzy graphs (FGs). In the beginning, we introduce several types of strong arcs in IFDGs, like semi-β strong arcs, semi-δ strong arcs, etc. Then, we introduce the concepts of domination in IFDGs based on these strong arcs and discuss its various useful characteristics. Moreover, the dominating set (DS), minimal dominating set (MDS), etc., are described with some fascinating results. We also introduce the concept of an independent set in IFDGs and investigate its relations with the DS, minimal independent set (MIS) and MDS. We also provide numerous important characterizations of domination in IFDGs based on minimal and maximal dominating sets. In this context, we discuss the lower and upper dominations of some IFDGs. In addition, we introduce the terms status and structurally equivalent and examine a few relationships with the dominations in IFDGs. Finally, we investigate the most expert (influential) person in the organization by utilizing the concepts of domination in IFGs.

After introducing and developing fuzzy graph theory, a lot of studies have been done in this field. The object of this paper is to demonstrate various Dominations such as Edge domination, Total domination, Strong (weak) domination, Regular domination, connected domination, Split (non-split) domination in fuzzy graphs with their importance and applications in real world. We explored about Inverse Dominations in Fuzzy graphs. Some results are derived to various dominations in Fuzzy graphs. Fuzzy graphs found an increasing number of applications in prevailing science where the information inherent in the system varies with different levels of precision. We prompt some applications in modeling traffic and transportation problems, telecommunications, job allocation and at ATM centers. The wide varieties of domination parameters are defined.

Fuzzy graph (FG) models take on the presence being ubiquitous in environmental and fabricated structures by humans, specifically the vibrant processes in physical, biological, and social systems. Owing to the unpredictable and indiscriminate data which are intrinsic in real life and problems being often ambiguous, it is very challenging for an expert to exemplify these problems through applying an FG. Vague graph structure (VGS), belonging to the FG family, has good capabilities when facing with problems that cannot be expressed by FGs. VGS can handle the vagueness connected with the incompatible and determinate information of any real-world problem, where FGs may not succeed to bear satisfactory results. The previous definitions’ restrictions in FGs have led us to propose new definitions in the VGS. Domination is one of the key issues that has many applications in computer science and social networks. Today, many researchers are trying to prove its application in medical sciences and psychology. Therefore, in this paper, different concepts related to domination in VGSs such as Ni-dominating set, vague full dominating set, minimal N i -DS, and strong capacity Ni-dominating set are defined using some examples. Finally, an application of domination in medical sciences has been presented.

An application of fuzzy graph in accidental prone zone to reduce the traffic congestion

In this paper, a fuzzifying matroid is induced respectively from a fuzzy graph and a fuzzy vector subspace. The concepts of graphic fuzzifying matroid and representable fuzzifying matroid are presented and some properties of them are discussed. In general, a graphic fuzzifying matriod can not be representable over any field. But when a fuzzifying matroid is isomorphic to a fuzzifying cycle matroid which is induced by a fuzzy tree, it is a representable over any field.