Energy of inverse graphs of dihedral and symmetric groups

O Ejima,K O Aremu,A Audu

doi:10.1186/s42787-020-00101-8

O Ejima, K O Aremu + Show 1 more

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https://doi.org/10.1186/s42787-020-00101-8

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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}^{+}).

Highlights

Graph representation is one of the combinatorial properties used to understand some interesting properties of complex structures
In this paper, motivated by the works of Alfuraidan and Zakariya [1], Gutman [7], and Fadzil et al [9], we compute the energy of some inverse graphs which is the sum of the absolute values of the eigenvalues of adjacency matrices of their corresponding inverse graphs of finite groups
We study the energy of dihedral and symmetry groups and show that if G is a finite non-abelian group with exactly two non-self invertible elements, the associated inverse graph (G) is never a complete bipartite graph

Summary

Introduction

Graph representation is one of the combinatorial properties used to understand some interesting properties of complex structures. Alfuraidan and Zakariya [1] introduced and studied the inverse graphs associated with finite groups. In this paper, motivated by the works of Alfuraidan and Zakariya [1], Gutman [7], and Fadzil et al [9], we compute the energy of some inverse graphs which is the sum of the absolute values of the eigenvalues of adjacency matrices of their corresponding inverse graphs of finite groups.

Results

Conclusion