Domination Number of Cayley Graphs on Finite Abelian Groups

In this paper, we study the domination parameters of Cayley graphs constructed out of Zp×Zm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_{p}\times {\mathbb {Z}}_{m}$$\end{document}, where m∈{pα,pαqβ,pαqβrγ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in \{p^{\alpha }, p^{\alpha }q^{\beta }, p^{\alpha }q^{\beta }r^{\gamma }\}$$\end{document} and p, q, r are prime numbers. Indeed, we give a lower bound for domination number of Γ=Cay(Z2×Z2αp1α1p2α2…pkαk,Φ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ={\mathrm {Cay}}({\mathbb {Z}}_{2}\times {\mathbb {Z}}_{2^{\alpha }p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\ldots p_{k}^{\alpha _{k}}},\Phi )$$\end{document}, where Φ=φ2×φ2αp1α1p2α2…pkαk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi =\varphi _{2}\times \varphi _{2^{\alpha }p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\ldots p_{k}^{\alpha _{k}}}$$\end{document} , p1=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{1}=3$$\end{document} and for 1≤i≤k-1,pi+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le i\le k-1,~p_{i+1}$$\end{document} is the first prime greater than pi,α≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{i},~\alpha \ge 2$$\end{document} and α1,α2…αk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{1}, \alpha _{2}\ldots \alpha _{k}$$\end{document} are positive integers, and φm={ℓ|1≤ℓ<m,gcd(ℓ,m)=1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{m}=\{\ell | 1\le \ell <m, \gcd (\ell , m)=1\}$$\end{document}.