On the signed domination number of some Cayley graphs

Abstract

A signed dominating function of graph $\Gamma$ is a function $g:V(\Gamma )⟶\{-1,1\}$ 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 $G=⟨S⟩$ be a finite group such that $e\notin S=S^{-1}$. 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 ${\gamma }_{{}_S}(Cay(S:G))$.

10 pages, 4 figures