Some isomorphic properties of m-polar fuzzy graphs with applications.

Ganesh Ghorai,Madhumangal Pal

doi:10.1186/s40064-016-3783-z

Abstract

The theory of graphs are very useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. In this paper, we present a frame work to handle m-polar fuzzy information by combining the theory of m-polar fuzzy sets with graphs. We introduce the notion of weak self complement m-polar fuzzy graphs and establish a necessary condition for m-polar fuzzy graph to be weak self complement. Some properties of self complement and weak self complement m-polar fuzzy graphs are discussed. The order, size, busy vertices and free vertices of an m-polar fuzzy graphs are also defined and proved that isomorphic m-polar fuzzy graphs have same order, size and degree. Also, we have presented some results of busy vertices in isomorphic and weak isomorphic m-polar fuzzy graphs. Finally, a relative study of complement and operations on m-polar fuzzy graphs have been made. Applications of m-polar fuzzy graph are also given at the end.

Highlights

After the introduction of fuzzy sets by Zadeh (1965), fuzzy set theory have been included in many research fields
The theory of fuzzy sets has become a vigorous area of research in different disciplines including medical and life sciences, management sciences, social sciences engineering, statistic, graph theory, artificial intelligence, signal processing, multi agent systems, decision making and automata theory
We have proved some results of busy vertices in isomorphic and weak isomorphic m-polar fuzzy graphs

Summary

Introduction

After the introduction of fuzzy sets by Zadeh (1965), fuzzy set theory have been included in many research fields. Definition 6 (Ghorai and Pal 2016a) The composition of two m-polar fuzzy graphs G1 = (V1, A1, B1) and G2 = (V2, A2, B2) of the graphs G1∗ = (V1, E1) and G2∗ = (V2, E2) respectively is denoted as a pair G1[G2] = (V1 × V2, A1 ◦ A2, B1 ◦ B2) such that for i = 1, 2, . Definition 7 (Ghorai and Pal 2016a) The union G1 ∪ G2 = (V1 ∪ V2, A1 ∪ A2, B1 ∪ B2) of the m-polar fuzzy graphs G1 = (V1, A1, B1) and G2 = (V2, A2, B2) of G1∗ and G2∗ respectively is defined as follows: for i = 1, 2, .

Results

Conclusion