Distance Laplacian spectra of joined union of graphs

The distance Laplacian matrix of a simple connected graph G is defined as D-L(C) = 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 determine the distance Laplacian spectra of the graphs obtained by generalization of the join and lexicographic product of graphs (namely joined union). It is shown that the distance Laplacian spectra of these graphs not only depend on the distance Laplacian spectra of the participating graphs but also depend on the spectrum of another matrix of vertex-weighted Laplacian kind (analogous to the definition given by Chung and Langlands [A combinatorial Laplacian with vertex weights, J. Combin. Theory Ser. A 75 (1996) 316-327]).