Group-annihilator graphs realised by finite abelian groups and its properties

Abstract

Let $G$ be a finite abelian group viewed a $\mathbb{Z}$-module and let $\mathcal{G}=(V,E)$ be a simple graph. In this paper, we consider a graph $\Gamma (G)$ called as a \textit{group-annihilator} graph. The vertices of $\Gamma (G)$ are all elements of $G$ and two distinct vertices $x$ and $y$ are adjacent in $\Gamma (G)$ if and only if $[x:G][y:G]G=\{0\}$, where $x,y\in G$ and $[x:G]=\{r\in \mathbb{Z}:rG\subseteq \mathbb{Z}x\}$ is an ideal of a ring $\mathbb{Z}$. We discuss in detail the graph structure realised by the 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-annhilator graph. We show that the Laplacian eigen values are representatives of orbits of the group action: $Aut(\Gamma (G))\times G\to G$.