Perfect codes and regular sets in vertex-transitive graphs

A subset $C$ of the vertex set $V$ of a graph $\mathrm{\Gamma }=(V,E)$ 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 $\text{Cos}(G,H,U)$. 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 $\text{Cos}(G,H,U)$ 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 $N_A(H)\mathrm{/}H$ being a perfect code of $(N_G(H)\mathrm{/}H,1_{N_G(H)\mathrm{/}H})$.