Perfect codes and regular sets in vertex-transitive graphs

Abstract: A subset ( C ) of the vertex set ( V ) of a graph ( Γ= (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}}) ).