Subgroup sum graphs of finite abelian groups

TL;DR

This paper investigates the structure and properties of subgroup sum graphs of finite abelian groups, including their variants, analyzing parameters like perfectness, clique number, and spectrum, with detailed focus on prime sum graphs.

Contribution

It provides a comprehensive analysis of subgroup sum graphs and their extended versions, including structural properties and spectral characteristics, especially for prime sum graphs.

Findings

Determined conditions for perfectness and connectedness.

Calculated clique, independence, and domination numbers.

Analyzed spectral properties of subgroup sum graphs.

Abstract

Let $G$ be a finite abelian group, written additively, and $H$ a subgroup of~$G$. The \emph{subgroup sum graph} $\Gamma_{G,H}$ is the graph with vertex set $G$, in which two distinct vertices $x$ and $y$ are joined if $x+y\in H\setminus\{0\}$. These graphs form a fairly large class of Cayley sum graphs. Among cases which have been considered previously are the \emph{prime sum graphs}, in the case where $H=pG$ for some prime number $p$. In this paper we present their structure and a detailed analysis of their properties. We also consider the simpler graph $\Gamma^+_{G,H}$, which we refer to as the \emph{extended subgroup sum graph}, in which $x$ and $y$ are joined if $x+y\in H$: the subgroup sum is obtained by removing from this graph the partial matching of edges having the form $\{x,-x\}$ when $2x\ne0$. We study perfectness, clique number and independence number, connectedness, diameter, spectrum, and domination number of these graphs and their complements. We interpret our general results in detail in the prime sum graphs.