On the Connectivity and Equality of Some Graphs on Finite Semigroups

Abstract

In this paper, we study various graphs, namely Colon power graph, cyclic graph, enhanced power graph, and commuting graph on a semigroup S. The purpose of this paper is twofold. First, we study the interconnection between the diameters of these graphs on semigroups having one idempotent. Consequently, the results on the connectedness and the diameter of the proper enhanced power graphs (or cyclic graph) of finite groups, viz., symmetric group and alternating group, are obtained. In the other part of this paper, for an arbitrary pair of these four graphs, we classify finite semigroups such that the graphs in this pair are equal. Our results generalize some of the corresponding results of these graphs on groups to semigroups.