On maximum degree (signless) Laplacian matrix of a graph

Let G be a simple graph on n vertices and v1, v2,...,vn be the vértices of G. We denote the degree of a vertex vi in G by dG(vi) = di. The maximum degree matrix of G, denoted by M(G), is the real symmetric matrix with its ijth entry equal to max{di, dj} if the vertices vi and vj are adjacent in G, 0 otherwise. In analogous to the definitions of Laplacian matrix and signless Laplacian matrix of a graph, we consider Laplacian and signless Laplacian for the maximum degree matrix, called the maximum degree Laplacian matrix and the maximum degree signless Laplacian matrix, respectively. Also, we introduce maximum degree Laplacian energy and maximum degree signless Laplacian energy of a graph. Then we determine the maximum degree (signless) Laplacian energy of some graphs in terms of ordinary energy, and (signless) Laplacian energy. We compute the máximum degree (signless) Laplacian spectra of some graph compositions. A lower and upper bound for the largest eigenvalue of the maximum degree (signless) Laplacian matrix is established and also we determine an upper bound for the second smallest eigenvalue of maximum degree Laplacian matrix in terms of vertex connectivity. We also determine bounds for the maximum degree (signless) Laplacian energy in terms of first Zagreb index.