Energy of inverse graphs of dihedral and symmetric groups

Abstract

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 D p of the dihedral group D n of order 2p and subgroup S k of the symmetric groups S n of order k! such that 2 p = n ! ( p , n , k ≥ 3 and p , n , k ∈ ℤ + ) . $2p = n!~(p,n,k \geq 3~\text {and}~p,n,k \in \mathbb {Z}^{+}).$

Keywords

• Inverse graphs
• Energy of graphs
• Eigenvalues
• Adjacency matrix
• Finite groups
• Graph of finite groups