Regular sets in Cayley sum graphs on generalized dicyclic groups
Meiqi Peng, Yuefeng Yang
TL;DR
This work classifies $(\alpha,\beta)$-regular sets arising from Cayley sum graphs on the generalized dicyclic group $G$, focusing on all subgroups $H\le A$ and the two- coset form $\langle H,zb\rangle$. Using the abelian structure of $A$, the involution $b$, and a detailed coset analysis, the authors derive explicit necessary and sufficient conditions for $(\alpha,\beta)$ to occur, expressed in terms of invariants $|L|$, $m$, $r$, and the membership of $B$ and $b^2$ in $H$. For $H$, they show $H$ is $(\alpha,\beta)$-regular iff $H$ is $(\alpha,0)$-regular and $(0,\beta)$-regular, with precise bounds and parity constraints; for $\langle H,zb\rangle$, they provide a decomposition-based criterion tying regularity to the regularity of $S\cap A$ and $S\cap Ab$. Collectively, the results extend the theory of subgroup regular codes in Cayley sum graphs to generalized dicyclic groups, enabling constructive design of regular sets with specified adjacency patterns. The findings have potential implications for combinatorial designs and network coding within algebraically structured graphs.
Abstract
For a graph $Γ=(V(Γ),E(Γ))$, a subset $C$ of $V(Γ)$ is called an $(α,β)$-regular set in $Γ$, if every vertex of $C$ is adjacent to exactly $α$ vertices of $C$ and every vertex of $V(Γ)\setminus C$ is adjacent to exactly $β$ vertices of $C$. In particular, if $C$ is an $(α,β)$-regular set in some Cayley sum graph of a finite group $G$ with connection set $S$, then $C$ is called an $(α,β)$-regular set of $G$. In this paper, we consider a generalized dicyclic group $G$ and for each subgroup $H$ of $G$, by giving an appropriate connection set $S$, we determine each possibility for $(α,β)$ such that $H$ is an $(α,β)$-regular set of $G$.