  The search for a highly discriminating and easily computable invariant to
distinguish graphs remains a challenging research topic. Here we focus on
cospectral graphs whose complements are also cospectral (generalized
cospectral), and on coinvariant graphs (same Smith normal form) whose
complements are also coinvariant (generalized coinvariant). We show a new
characterization of generalized cospectral graphs in terms of codeterminantal
graphs. We also establish the Smith normal form of some graph classes for
certain associated matrices, and as an application, we prove that the Smith
normal form can be used to uniquely determine star graphs. Finally, for graphs
up to 10 vertices, we present enumeration results on the number of generalized
cospectral graphs and generalized coinvariant graphs with respect to several
associated matrices.
