Determinants of Box Products of Paths

Abstract

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. If n=m, H.M. Rara showed in 1996 that det(M)=0. We extend this result to allow n and m to be any positive integers, and show that, if gcd(n+1,m+1)>1, then det(M)=0; otherwise, if gcd(n+1,m+1)=1, then det(M)=(-1)^(nm/2).