  We study the distribution of the cokernels of adjacency matrices (the Smith
groups) of certain models of random $r$-regular graphs and directed graphs,
using recent mixing results of M\'esz\'aros. We explain how convergence of such
distributions to a limiting probability distribution implies asymptotic
nonsingularity of the matrices, giving another perspective on recent results of
Huang and M\'esz\'aros on asymptotic nonsingularity of adjacency matrices of
random regular directed and undirected graphs, respectively. We also remark on
the new distributions on finite abelian groups that arise, in particular in the
$p$-group aspect when $p\mid r$.

Probability;Combinatorics