2025/12/24 by Aliréza Abdollahi, Abdollahi, Alireza, Zeinab Akhlaghi +3
Mathematics · Engineering · #Finite Group Theory Research #Rings, Modules, and Algebras #graph theory and CDMA systems
paper · doi:10.48550/arxiv.2512.21242
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 \( NA(H)/H \) being a perfect code of \(( NG(H)/H, 1_NG(H)/H) \).