On distance signless Laplacian eigenvalues of zero divisor graph of commutative rings

For a simple connected graph $G$ of order $n$, the distance signless Laplacian matrix is defined by $D^Q(G)=D(G)+Tr(G)$, where $D(G)$ and $Tr(G)$ is the distance matrix and the diagonal matrix of vertex transmission degrees, respectively. The zero divisor graph $\mathrm{\Gamma }(R)$ of a finite commutative ring $R$ is a simple graph, whose vertex set is the set of non-zero zero divisors of $R$ and two vertices $v,w\in \mathrm{\Gamma }(R)$ are edge connected whenever $vw=wv=0$. In this article, we find the $D^Q$-eigenvalues of zero divisor graph of the ring ${\mathbb{Z}}_n$ for general value $n=p_1^{l_1}{p}_2^{l_2}$, where $p_1<p_2$ are distinct prime numbers and $l_1,l_2\in \mathbb{N}$. Further, we investigate the $D^Q$-eigenvalues of zero divisor graphs of local rings and the rings whose associated zero divisor graphs are Hamiltonian. Also, we obtain the trace norm and the Wiener index of $\mathrm{\Gamma }({\mathbb{Z}}_n)$ for some special values of $n$.