On signless Laplacian spectrum of the zero divisor graphs of the ring $\mathbb{Z}_{n}$

Abstract

For a finite commutative ring $ R $ with identity $ 1\neq 0 $, the zero divisor graph $ \Gamma(R) $ is a simple connected graph having vertex set as the set of nonzero zero divisors of $ R $, where two vertices $ x $ and $ y $ are adjacent if and only if $ xy=0$. We find the signless Laplacian spectrum of the zero divisor graphs $ \Gamma(\mathbb{Z}_{n}) $ for various values of $ n$. Also, we find signless Laplacian spectrum of $ \Gamma(\mathbb{Z}_{n}) $ for $ n=p^z , z\geq 2 $, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise $ n $ for which zero divisor graph $ \Gamma(\mathbb{Z}_{n}) $ are signless Laplacian integral.