Determinants of Box Products of Paths

TL;DR

This paper extends previous results on the determinants of adjacency matrices of box products of paths, establishing conditions based on the greatest common divisor of (n+1) and (m+1) for when the determinant is zero or non-zero.

Contribution

The authors generalize the determinant result for the adjacency matrix of box products of paths to all positive integers n and m, based on gcd conditions.

Findings

det(M)=0 if gcd(n+1,m+1)>1

det(M)=(-1)^(nm/2) if gcd(n+1,m+1)=1

extends prior work from equal path lengths to all positive integers

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).