Cayley fuzzy graphs on the fuzzy groups

Abstract

Given non-trivial group G and a non-empty subset S of G such that not containing the identity element of G and \(S = S^{-1} := \{s^{-1} \mid s \in S\}\), the Cayley graph \(\varGamma = Cay(G, S)\) is the graph whose vertex set \(V(\varGamma )\) is G, and edge set \(E(\varGamma )\) is \(\{\{g, sg\} \mid g \in G, s \in S\}\). The Cayley graphs are presently being considered as models architecture for interconnection networks. When there is an uncertainty on vertices or edges, fuzzy graph has special importance. In this paper, we develop the concept of Cayley fuzzy graphs on the fuzzy groups and investigate some basic properties of Cayley fuzzy graphs. We introduce some sufficient conditions on the fuzzy graphs that under it’s, a fuzzy graph is a Cayley fuzzy graph and discuss some operations of the fuzzy graphs on the Cayley fuzzy graphs, prove that the product of two Cayley fuzzy graphs is a Cayley fuzzy graph, for some products. Also we provide an application of Cayley fuzzy graphs, and use Dijkstra’s Algorithm for finding the fuzzy shortest path in a Cayley fuzzy graph.