Group-Annihilator Graphs Realised by Finite Abelian Groups and Its Properties

Abstract

Let G be a finite abelian group viewed a Z-module and let G= (V, E) be a simple graph. In this paper, we consider a graph Γ (G) called as a group-annihilator graph. The vertices of Γ (G) are all elements of G and two distinct vertices x and y are adjacent in Γ (G) if and only if [x: G] [y: G] G= { 0 } , where x, y∈ G and [x: G] = { r∈ Z: rG⊆ Zx} is an ideal of a ring Z. We discuss in detail the graph structure realised by a group G. Moreover, we study the creation sequence, hyperenergeticity and hypoenergeticity of group-annihilator graphs. Finally, we conclude the paper with a discussion on Laplacian eigen values of the group-annihilator graph. We show that the Laplacian eigen values are representatives of orbits of the group action: Aut(Γ (G)) × G→ G.