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

TL;DR

This paper introduces the group-annihilator graph for finite abelian groups, analyzing its structure, spectral properties, and energy characteristics, revealing connections between group actions and graph eigenvalues.

Contribution

It defines the group-annihilator graph for finite abelian groups and investigates its structural, energetic, and spectral properties, including Laplacian eigenvalues related to group automorphisms.

Findings

The Laplacian eigenvalues correspond to orbits under the automorphism group action.

The graph's structure is explicitly characterized for various finite abelian groups.

Properties like hyperenergeticity and hypoenergeticity are studied in detail.

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