2023/08/22 by Asamin Khaefi, Khaefi, Asamin, Zeinab Akhlaghi +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Immunology and Microbiology · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Macrophage Migration Inhibitory Factor #Protein Tyrosine Phosphatases
paper · pdf · doi:10.48550/arxiv.2308.11434
openalex publication_date 2023/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A subset C of the vertex set of a graph Γ is said to be (a,b)-regular if C induces an a-regular subgraph and every vertex outside C is adjacent to exactly b vertices in C. In particular, if C is an (a,b)-regular set of some Cayley graph on a finite group G, then C is called an (a,b)-regular set of G and a (0,1)-regular set is called a perfect code of G. In [Wang, Xia and Zhou, Regular sets in Cayley graphs, J. Algebr. Comb., 2022] it is proved that if H is a normal subgroup of G, then H is a perfect code of G if and only if it is an (a,b)-regular set of G, for each 0≤ a≤|H|-1 and 0≤ b≤|H| with gcd(2,|H|-1)| a. In this paper, we generalize this result and show that a subgroup H of G is a perfect code of G if and only if it is an (a,b)-regular set of G, for each 0≤ a≤|H|-1 and 0≤ b≤|H| such that gcd(2,|H|-1) divides a.