On the signless Laplacian and normalized Laplacian spectrum of the zero divisor graphs

Let R be a commutative ring with nonzero identity and let Γ(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma (R)$$\end{document} denote the zero divisor graph of R. In this paper, we describe the signless Laplacian and normalized Laplacian spectrum of the zero divisor graph Γ(Zn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma (\mathbb {Z}_n)$$\end{document}, and we determine these spectrums for some values of n. We also characterize the cases that 0 is a signless Laplacian eigenvalue of Γ(Zn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma (\mathbb {Z}_n)$$\end{document}. Moreover, we find some bounds for some eigenvalues of the signless Laplacian and normalized Laplacian matrices of Γ(Zn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma (\mathbb {Z}_n)$$\end{document}.