We introduce a Laplacian for the distance matrix of a connected graph, called the distance Laplacian and we study its spectrum. We show the equivalence between the distance Laplacian spectrum and the distance spectrum 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. Finally, we investigate some particular distance Laplacian eigenvalues.
Published December 2011 , 15 pages