On subgroup perfect codes in Cayley sum graphs

TL;DR

This paper investigates the conditions under which subgroups of finite groups serve as perfect or total perfect codes in Cayley sum graphs, providing classifications for certain group families.

Contribution

It establishes necessary conditions for subgroups to be perfect codes in Cayley sum graphs and classifies such graphs for specific group families.

Findings

Identifies necessary conditions for subgroups to be perfect codes

Classifies Cayley sum graphs with subgroup perfect codes for certain groups

Provides a framework for understanding subgroup codes in algebraic graph structures

Abstract

A perfect code $C$ in a graph $\Gamma$ is an independent set of vertices of $\Gamma$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code $C$ in $\Gamma$ is a set of vertices of $\Gamma$ such that every vertex of $\Gamma$ is adjacent to a unique vertex in $C$. Let $G$ be a finite group and $X$ a normal subset of $G$. The Cayley sum graph $\mathrm{CS}(G,X)$ of $G$ with the connection set $X$ is the graph with vertex set $G$ and two vertices $g$ and $h$ being adjacent if and only if $gh\in X$ and $g\neq h$. In this paper, we give some necessary conditions of a subgroup of a given group being a (total) perfect code in a Cayley sum graph of the group. As applications, the Cayley sum graphs of some families of groups which admit a subgroup as a (total) perfect code are classified.