Spectral distances in some sets of graphs

Some of the spectral distance related parameters (cospectrality, spectral eccentricity, and spectral diameter with respect to an arbitrary graph matrix) are determined in one particular set of graphs. According to these results, the spectral distances connected with the adjacency matrix and the corresponding distance related parameters are computed in some sets of trees. Examples are provided of graphs whose spectral distances related to the adjacency matrix, the Laplacian and the signless Laplacian matrix are mutually equal. The conjecture related to the spectral diameter of the set of connected regular graphs with respect to the adjacency matrix is disproved using graph energy.