  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 (a
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
present 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 to distinguish graphs in cases where other algebraic
invariants, such as those derived from the spectrum, fail. Finally, we show a
new graph characterization using the Smith normal form of the signless distance
Laplacian matrix.
