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

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 $\mathrm{\Gamma }(G)$ called as a $\mathit{\text{group-annihilator}}$ graph. The vertices of $\mathrm{\Gamma }(G)$ are all elements of $G$ and two distinct vertices $x$ and $y$ are adjacent in $\mathrm{\Gamma }(G)$ 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: $Aut(\mathrm{\Gamma }(G))\times G\to G$.