On the signed domination number of some Cayley graphs

TL;DR

This paper investigates the signed domination number of Cayley graphs generated by subsets of finite groups, providing formulas based on subset size and classifying groups accordingly.

Contribution

It derives the signed domination number for Cayley graphs from finite groups and classifies groups based on subset size and domination number.

Findings

Signed domination number formulas depending on subset size

Classification of groups based on domination number and subset size

Explicit determination of domination number for specific Cayley graphs

Abstract

A signed dominating function of graph $\Gamma$ is a function $g :V(\Gamma) \longrightarrow \{-1,1\}$ such that $\sum_{u \in N[v]}g(u) >0$ for each $v \in V(\Gamma)$. The signed domination number $\gamma_{_S}(\Gamma)$ is the minimum weight of a signed dominating function on $\Gamma$. Let $G=\langle S \rangle$ be a finite group such that $e \not\in S=S^{-1}$. In this paper, we obtain the signed domination number of $Cay(S:G)$ based on cardinality of $S$. Also we determine the classification of group $G$ by $|S|$ and $\gamma_{_S}(Cay(S:G))$.