On distance Laplacian spectrum energy of graphs

For a simple connected graph G of order n having distance Laplacian eigenvalues rho(L)(1) >= rho(L)(2) >= ... >= rho(L)(n), the distance Laplacian energy DLE(G) is defined as DLE(G) = Sigma(n)(i=1) vertical bar rho(L)(i) - 2W(G)/n vertical bar, where W(G) is the Wiener index of G. We obtain the distance Laplacian spectrum of the joined union of graphs G1, G2, ..., G(n) in terms of their distance Laplacian spectrum and the spectrum of an auxiliary matrix. As application, we obtain the distance Laplacian spectrum of the lexicographic product of graphs. We study the distance Laplacian energy of connected graphs with given chromatic number X. We show that among all connected graphs with chromatic number x the complete X-partite graph has the minimum distance Laplacian energy. Further, we discuss the distribution of distance Laplacian eigenvalues around average transmission degree 2W(G)/n.