Characteristics of Fuzzy Petersen Graph and Platonic Graph with Fuzzy Rule

Abstract

Graph theory is the concepts used to study and model various application in different areas. In this paper, we consider the The Petersen and also the Platonic graph using if-then-rules fuzzy numbers. The results are related to the find the degree of odd vertices and even verticesare same by applying if-then-rules through the paths described by fuzzy numbers.

Introduction

I. INTRODUCTION

The first definition of fuzzy graph was introduced by Kaufmann (1973), based on Zadeh’s fuzzy relations (1971) A more elaborate definition is due to Azriel Rosenfeld who considered fuzzy relation on fuzzy sets and developed the theory of fuzzy graph in 1975.

During the same time Yeh and Bang have also introduced various connectedness concepts in fuzzy graph. Till now fuzzy graphs has been witnessing a tremendous growth and finds applications in many branches of engineering and technology.

Fuzzy systems based on fuzzy if-then rules have been successfully applied to varioustheorems in the field of fuzzy control. Fuzzy Rule based system has high comprehensibility because human users can easily understand the meaning of each fuzzy if-then rule through its linguistic interpretation. Graph theory has numerous applications to problems in system analysis, operation research, transportation and economics. In many cases, however some aspects of graph theoretic problem may be uncertain. For example, the vehicle travel time or vehicle capacity on a road network may not be known exactly. In such cases, it is natural to deal with the uncertainly using fuzzy set theory. The concepts of a fuzzy graph are a natural generalization of crisp graphs, given by Rosenfeld [Zadeh et al., 1975] .

II. BASIC CONCEPT

A. Definition 2.1

A graph G consists of a pair G: (V,E) where V(G) is a non empty finite set whose elements are called points or vertices and E(G) is a set of unordered pairs of distinct elements of V(G). The elements of E(G) are called lines or edges of the graph G.

III. FUZZY PETERSEN GRAPH WITH FUZZY RULE

Conclusion

In this paper we obtain that petersen graph and platonic graph . We conclude that the odd degree of vertices is equal to the even degree of vertices, for both the above cases with the help of IF-THEN rules (i.e., L.H.S=R.H.S).Finally, IF-THEN rules applied through shortest paths are shown with different analysis.

References

[1] Dr.G.Nirmala,P.Sindhamani “Characteristics of fuzzy Petersen graph with fuzzy rule”,journal,2014. [2] Dr.Dr.G.Nirmala,S.Prabavathi “Application of fuzzy if-then rule in Peterson graph”, volume 4,issue 8,August 2014. [3] M.Tamilarasi,Dr.S.S.Dhenakaran “Hamiltonian Approach for shortest path”,IJSRT 184894 June 2018. [4] Sunny Dagar,”Modified prim’s Algorithm” volume 03, volume 02,2012. [5] Dr.DVijayalakshmi,R.Kalaivani “Minimum cost spanning tree using matrix algorithm “Journal.

Copyright

Copyright © 2022 Suguna. R, Naveen. L. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.