CODEN(USA): JSERBR On some properties combinatorics of Graphs in the d-dimensional FLS (original) (raw)

Recent Developments on the Basics of Fuzzy Graph Theory

Handbook of Research on Advanced Applications of Graph Theory in Modern Society

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.

Linear fuzzy graphs

Fuzzy Sets and Systems - FSS, 1983

We introduce the concept of linear fuzzy graph and give its matrix and functional representations. This allows a generalization of the notions of semiorder and pseudo-order.

A NEW APPROACH ON REGULAR FUZZY GRAPH

1.1 Graph Graph theory is tremendously useful in modelling the essential features of systems with finite components. Wide applications of graphical models are in the field of railway network, telephone network, communication problems, traffic network etc. Graph theoretic models can sometimes provide a useful structure upon which analytic techniques can be used. A graph is also used to model a relationship between a given set of objects. Each object is represented by a vertex and the relationship between them is represented by an edge if the relationship is unordered and by means of a directed edge if the objects have an ordered relation between them. Relationship among the objects need not always be precisely defined criteria; when we think of an imprecise concept, the fuzziness arises. 1.2 Fuzzy graph A mathematical frame work to explain the concept of uncertainty in real life through the publication of a seminal paper is introduced by Zadeh [1]. A fuzzy set is defined mathematically by assigning to each possible individual in the universe of discourse a value, representing its grade of membership, which corresponds to the degree, to which that individual is similar or compatible with the concept represented by the fuzzy set. Rosenfeld [2] was introduced the fuzzy graph using fuzzy relation, represents the relationship between the objects by precisely indicating the level of the relationship between the objects of the given set. Also he coined many fuzzy analogous graph theoretic concepts like bridge, cut vertex and tree. Fuzzy graphs have many more applications in modelling real time systems where the level of information inherent in the system varies with different levels of precision.

Some Properties on Fuzzy Graphs

In this study researchers propose the complete, regular and complement fuzzy graph. In these subjects we shall study some properties such that: self complementary, regular, total regular fuzzy graph and the relation between these subject. On the other hand we shall study a product fuzzy graph and what the relation between the product and the complete also the complement. We get some theorem, proposition and corollaries for these subjects.

Completeness and regularity of generalized fuzzy graphs

SpringerPlus, 2016

Background Nowadays, graphs do not represent all the systems like networks, routes, schedules, images, etc. properly due to the uncertainty or haziness of the parameters of systems. For example, a social network may be represented as a graph, where vertices represent an account (person, institution, etc.) and edges represent the relation between those accounts. If the relations among accounts are measured as either good or bad according to the frequency of contacts among those accounts, then fuzzyness can be added for such representations. This and many other problems lead to define fuzzy graphs. The first definition of a fuzzy graph was introduced by Kauffman (1973). But, Rosenfeld (1975) described fuzzy relations on fuzzy sets and developed some theory of fuzzy graphs. Using these concept of fuzzy graphs, Koczy (1992) discussed fuzzy graphs to evaluate and to optimize any networks. Samanta and Pal (2013) showed that fuzzy graphs can be used in competition in ecosystems. After that, they introduced some different types of fuzzy graphs (Samanta and Pal 2015; Samanta et al. 2014). Bhutani and Battou (2003) and Bhutani and Rosenfeld (2003) discussed different arcs in fuzzy graphs. For further details of fuzzy graphs, readers may look in Mathew (2009), Mordeson and Nair (2000), Pramanik et al. (2014, 2016) and Rashmanlou et al. (2015). Applications of fuzzy graph include data mining, image segmentation, clustering, image capturing, networking, communication, planning, scheduling, etc.

Some new results on fuzzy graphs

2016 IEEE Region 10 Humanitarian Technology Conference (R10-HTC), 2016

We proposed new analogous results for fuzzy spanning sub graphs, complete, simple, regular and connected fuzzy graphs along with proof. As far as we know, there are no such results available in the previous work.

Semitotal Blocks in Fuzzy Graphs

This paper is a study of semitotal blocks in fuzzy graphs. During the study some interesting results regarding the semitotal blocks in fuzzy graphs are obtained. It is observed that when 'B' is a block of a given fuzzy graph G:(V, σ, µ), then degree of the vertex B in semi total block fuzzy graph T STB F(G) is equal to the sum of the membership grade of the vertices in that block and the number of edges in T STB F(G) related to block B is V(B) with membership grade minimum of

Topological numbers of fuzzy soft graphs and their application

Information Sciences, 2024

The diagram kind of a graph is used to show accumulated data. Graphs can be utilized for a variety of purposes because this data can be either quantitative or qualitative. Graphs can be used to model different relationships and processes in physical, biological, and social media marketing systems, and in finding directions on a map. A graph with properties attached to its nodes and edges that emphasize its applicability to real-world systems is sometimes called a network. The idea of fuzzy sets has developed in numerous ways and across many areas since its establishment in 1965. Applications of this theory can be found in numerous fields for instance in recognition of patterns, management science, AI, computer science, medicine, also in control engineering. The progress of mathematics has reached a very high level and continues now. While classical graph theory is widely applied in several domains, there are instances where its outcomes can be subject to uncertainty. In order to address this challenge, the utilization of the fuzzy theory of graphs is adopted, as it offers more accurate outcomes. There is a lack of a parameterization tool in fuzzy graph theory, as a consequence Molodtsov introduced soft set theory, which is a rather recent way to talk about ambiguity and vagueness. It is becoming more and more popular among scholars and is a novel approach to uncertainty and ambiguity simulation. The concept of soft graphs offers a parameterized perspective on graphs. In this article, we defined some familiar graph families in a fuzzy soft (FS) environment and by calculating their degrees, derived important results for two versions of Sombor numbers. In the end, we discussed an application of calculated results and by comparison, checked the efficiency of Sombor numbers in a FS framework.

On Graph Structures in Fuzzy Environment Using Optimization Parameter

IEEE Access

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 mµ k-regular FGSs are described, where m ∈]0, 1] represents the degree of all vertices in G under mapping µ 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. INDEX TERMS Fuzzy graph structure, weighted mean product, vertex degree, vertex total degree, regular fuzzy graph structure.