Determinants of Box Products of Paths

Details Determinants of Box Products of Paths Determinants of Box Products of Paths

2011-10-16

arxiv.org/abs/1110.3497v1

Mathematics

Combinatorics

Scientific paper

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).

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