Cospectrality of graphs with respect to distance matrices

The distance, distance Laplacian and distance signless Laplacian spectra of a connected graph G are the spectra of the distance, distance Laplacian and distance signless Laplacian matrices of G. Two graphs are said to be cospectral with respect to the distance (resp. distance Laplacian or distance signless Laplacian) matrix if they share the same distance (resp. distance Laplacian or distance signless Laplacian) spectrum. If a graph G does not share its spectrum with any other graph, we say G is determined by its spectrum. In this paper we are interested in the cospectrality with respect to the three distance matrices. First, we report on a numerical study in which we looked into the spectra of the distance, distance Laplacian and distance signless Laplacian matrices of all the connected graphs on up to 10 vertices. Then, we prove some theoretical results about what we can deduce about a graph from these spectra. Among other results we identify some of the graphs determined by their distance Laplacian or distance signless Laplacian spectra.