Enumeration of cospectral and coinvariant graphs

We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (cospectral mate), or at least one other graph with the same Smith normal form (coinvariant mate) with respect to several matrices associated to a graph. The presented numerical data give some indication that possibly the Smith normal form of the distance Laplacian and the signless distance Laplacian matrices could be a finer invariant than the spectrum to distinguish graphs. Finally, we prove a graph characterization using the Smith normal form of the distance signless Laplacian matrix.We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (cospectral mate), or at least one other graph with the same Smith normal form (coinvariant mate) with respect to several matrices associated to a graph. The presented numerical data give some indication that possibly the Smith normal form of the distance Laplacian and the signless distance Laplacian matrices could be a finer invariant than the spectrum to distinguish graphs. Finally, we prove a graph characterization using the Smith normal form of the distance signless Laplacian matrix.A