  Let $R_{n}$ denote the graph with vertex set consisting of the squares of an
$n \times n$ grid, with two squares of the grid adjacent when they lie in the
same row or column. This is the square rook's graph, and can also be thought of
as the Cartesian product of two complete graphs of order $n$, or the line graph
of the complete bipartite graph $K_{n,n}$. In this paper we compute the Smith
group and critical group of the graph $R_{n}$ and its complement. This is
equivalent to determining the Smith normal form of both the adjacency and
Laplacian matrix of each of these graphs. In doing so we verify a 1986
conjecture of Rushanan.
