Spectrum of graphs obtained by operations

The distance signless Laplacian matrix of a simple connected graph G is defined as DQ(G) = Tr(G) + D(G), where D(G) is the distance matrix of G and Tr(G) is the diagonal matrix whose main diagonal entries are the vertex transmissions in G. In this paper, we first determine the distance signless Laplacian spectrum of the graphs obtained by generalization of the join and lexicographic product graph operations (namely joined union) in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix, determined by the graph G. As an application, we show that new pairs of auxiliary equienergetic graphs can be constructed by joined union of regular graphs.