  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: $Aut(\Gamma(G)) \times G \rightarrow G$.
