  It is well known that the spectrum and the Smith normal form of a matrix can
be computed in polynomial time. Thus, it is interesting to explore how good are
these parameters for distinguishing graphs. This is relevant since it is
related to the Graph Isomorphism Problem (GIP), which asks to determine whether
two graphs are isomorphic. In this paper, we explore the computational
advantages of using the spectrum and the Smith normal form of two matrices
associated with a graph. By considering the SNF or the spectrum of two matrices
of a graph as a single parameter, we compute the number of non-isomorphic
graphs with the same parameter with up to 9 vertices, and with up to 10
vertices when the number of graphs with 9 vertices with the same parameter is
less than 1000. Focusing on the best 20 combinations of matrices for graphs
with 10 vertices, we notice that the number of such graphs with a mate is less
than 100 in any of these 20 cases. This computational result improves on
similar previous explorations. Therefore, the use of the spectrum or the SNF of
two matrices at the same time shows a substantial improvement in distinguishing
graphs.
