Perfect codes and regular sets in vertex-transitive graphs

TL;DR

This paper investigates the structure of regular sets, including perfect codes, in vertex-transitive graphs, providing conditions for their existence and characterizing their algebraic properties via group and subgroup relationships.

Contribution

It establishes necessary and sufficient conditions for normal subgroups to be regular sets and links perfect codes in graphs to quotient group properties.

Findings

Characterization of regular sets in vertex-transitive graphs.

Conditions for normal subgroups to be regular sets.

Relationship between perfect codes and quotient groups.

Abstract

A subset \( C \) of the vertex set \( V \) of a graph \( \Gamma = (V,E) \) is termed an $(r,s)$-regular set if each vertex in \( C \) is adjacent to exactly \( r \) other vertices in \( C \), while each vertex not in \( C \) is adjacent to precisely \( s \) vertices in \( C \). A specific case, known as a $(0,1)$-regular set, is referred to as a perfect code. In this paper, we will delve into $(r,s)$-regular sets in the context of vertex-transitive graphs. It is noteworthy that any vertex-transitive graph can be represented as a coset graph \( \Cos(G,H,U) \). When examining a group \( G \) and a subgroup \( H \) of \( G \), a subgroup \( A \) that encompasses \( H \) is identified as an $(r,s)$-regular set related to the pair \( (G,H) \) if there exists a coset graph \( \Cos(G,H,U) \) such that the set of left cosets of \( H \) in \( A \) forms an $(r,s)$-regular set within this graph. In this paper, we present both a necessary and sufficient condition for determining when a normal subgroup \( A \) that includes \( H \) as a normal subgroup qualifies as an $(r,s)$-regular set for the pair \( (G,H) \). Furthermore, if \( A \) is a normal subgroup of \( G \) containing \( H \), we establish a relationship between \( A \) being a perfect code of \( (G,H) \) and the quotient \( N_A(H)/H \) being a perfect code of \(( N_G(H)/H, {1_{N_{G}(H)/H}}) \).