Energy of inverse graphs of dihedral and symmetric groups

Abstract

Let (G,∗) be a finite group and S={x∈G|x≠x−1} be a subset of G containing its non-self invertible elements. The inverse graph of G denoted by Γ(G) is a graph whose set of vertices coincides with G such that two distinct vertices x and y are adjacent if either x∗y∈S or y∗x∈S. In this paper, we study the energy of the dihedral and symmetric groups, we show that if G is a finite non-abelian group with exactly two non-self invertible elements, then the associated inverse graph Γ(G) is never a complete bipartite graph. Furthermore, we establish the isomorphism between the inverse graphs of a subgroup Dp of the dihedral group Dn of order 2p and subgroup Sk of the symmetric groups Sn of order k! such that 2p = n!~(p,n,k geq 3~text {and}~p,n,k in mathbb {Z}^{+}).