Some properties of the distance Laplacian eigenvalues of a graph

The distance Laplacian of a connected graph G is defined by (equation presented), where (equation presented) is the distance matrix of G, and Diag(Tr) is the diagonal matrix whose main entries are the vertex transmissions in G. The spectrum of (equation presented) is called the distance Laplacian spectrum of G. In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance Laplacian spectrum that enable us to derive the distance Laplacian characteristic polynomials for several classes of graphs.