Some properties of various graphs associated with finite groups

Abstract

In this paper we investigate some properties of the power graph and  commuting graph associated with a finite group, using their tree-numbers. Among other things, it is shown that the simple group \(L_2(7)\) can be characterized through the tree-number of its power graph.  Moreover, the classification of groups with power-free decomposition is presented. Finally,  we  obtain an explicit formula concerning the tree-number of commuting graphs associated with the Suzuki simple groups.