Set Adjacency Measures in Fuzzy Graphs

Abstract

The concept of adjacency of one vertex to another in a graph is extended to adjacency between any two subsets of vertices. Alternative set adjacency measures are suggested, all of which are based on the given vertex to vertex adjacency. Several specially featured subsets of vertices, some known (as clique, kernel, etc.) and other new (deterrent, communion and more), are defined by the proposed adjacency measures. The framework for the work is a fuzzy graph, where the adjacency level between two vertices is not restricted to zero and one, but may assume any value within the unit interval. The extensions of the paper axe therefore twofold. From the vertex adjacency to the more general notion of set adjacency measures, and from the ordinary, nonfuzzy graph, to the broader family of fuzzy graphs.