On distance signless Laplacian spectrum of graphs and spectrum of zero divisor graphs of Zn

For a simple connected graph G of order n, we obtain the distance signless Laplacian spectrum of the joined union of regular graphs G(1), G(2),..., G(n) in terms of their adjacency spectrum and the spectrum of an auxiliary matrix. As a consequence, we obtain the distance signless Laplacian spectrum of the zero divisor graphs of finite commutative rings Z(n) for some values of n. We show that Gamma(Z(n)) is not in general distance signless Laplacian integral for n = p(z), where p is any prime and z = 2. Also, we find the spectrum of Gamma(Z(pz)) for certain values of z.