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

Abstract

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 $ \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 \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 $ \Gamma(\mathbb{Z}_{n}) $ for some special values of $ n $.

Keywords

• distance signless laplacian matrix
• zero divisor graphs
• commutative rings
• trace norm