Determinants of box products of paths

Suppose that G is the graph obtained by taking the box product of a path of length n and a path of length m . Let M be the adjacency matrix of G . In 1996, Rara showed that, if n = m , then det ( M ) = 0 . We extend this result to allow n and m to be any positive integers, and show that det ( M ) = { 0 if  gcd ( n + 1 , m + 1 ) ≠ 1 , ( − 1 ) n m / 2 if  gcd ( n + 1 , m + 1 ) = 1 .