Two Laplacians for the distance matrix of a graph

We introduce a Laplacian and a signless Laplacian for the distance matrix of a connected graph, called the distance Laplacian and distance signless Laplacian, respectively. We show the equivalence between the distance signless Laplacian, distance Laplacian and the distance spectra for the class of transmission regular graphs. There is also an equivalence between the Laplacian spectrum and the distance Laplacian spectrum of any connected graph of diameter 2. Similarities between n, as a distance Laplacian eigenvalue, and the algebraic connectivity are established.