On the signed domination number of some Cayley graphs

A signed dominating function of graph $\mathrm{\Gamma }$ is a function $g :V(\Gamma) \longrightarrow {-1,1}$suchthat$\sum_{u \in N[v]}g(u) >0$foreach$v \in V(\Gamma)$\mathrm{.}Thesigneddominationnumber$\gamma_{_S}(\Gamma)$ is the minimum weight of a signed dominating function on $\mathrm{\Gamma }$. Let $G=⟨S⟩$ be a finite group such that $e\in ̸S=S^{-1}$. 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 $\mathrm{\mid }S\mathrm{\mid }$ and ${\gamma }_{{}_S}(Cay(S:G))$.