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

Abstract

Let R be a commutative ring with nonzero identity and let $$\Gamma (R)$$ denote the zero divisor graph of R. In this paper, we describe the signless Laplacian and normalized Laplacian spectrum of the zero divisor graph $$\Gamma (\mathbb {Z}_n)$$, and we determine these spectrums for some values of n. We also characterize the cases that 0 is a signless Laplacian eigenvalue of $$\Gamma (\mathbb {Z}_n)$$. Moreover, we find some bounds for some eigenvalues of the signless Laplacian and normalized Laplacian matrices of $$\Gamma (\mathbb {Z}_n)$$.