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

Abstract

For a finite commutative ring $\mathbb{Z}_{n}$ with identity $1\neq 0$, the zero divisor graph $\Gamma(\mathbb{Z}_{n})$ is a simple connected graph having vertex set as the set of non-zero zero divisors, where two vertices $x$ and $y$ are adjacent if and only if $xy=0$. We find the distance Laplacian spectrum of the zero divisor graphs $\Gamma(\mathbb{Z}_{n})$ for different values of $n$. Also, we obtain the distance Laplacian spectrum of $\Gamma(\mathbb{Z}_{n})$ for $n=p^z$, $z\geq 2$, in terms of the Laplacian spectrum. As a consequence, we determine those $n$ for which zero divisor graph $\Gamma(\mathbb{Z}_{n})$ is distance Laplacian integral.