On the signed domination number of some Cayley graphs

Abstract

A signed dominating function of graph Γ is a function $g:V(\Gamma )\to \{-1,1\}$ 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 Γ. Let $G=〈S〉$ be a finite group such that $e\notin S=S^{-1}.$ In this paper, we obtain the signed domination number of $\mathit{Cay}(S:G)$ based on cardinality of S. Also we determine the classification of group G by $\left|S\right|$ and ${\gamma }_S(\mathit{Cay}(S:G)).$

Keywords:

• Cayley graph
• finite group
• signed domination number

AMS SUBJECT CLASSIFICATION:

• 05C50
• 05C25

Acknowledgment

The authors would like to express their gratitude to the referee for her/his helpful comments.

Additional information

Funding

The research of the second author was in part supported by a grant from Yazd University research council.