The kernel of the adjacency matrix of a rectangular mesh

Details The kernel of the adjacency matrix of a rectangular mesh The kernel of the adjacency matrix of a rectangular mesh

2002-01-22

arxiv.org/abs/math/0201211v1

Mathematics

Combinatorics

15 pages, 17 figures

Scientific paper

Given an m x n rectangular mesh, its adjacency matrix A, having only integer entries, may be interpreted as a map between vector spaces over an arbitrary field K. We describe the kernel of A: it is a direct sum of two natural subspaces whose dimensions are equal to $\lceil c/2 \rceil$ and $\lfloor c/2 \rfloor$, where c = gcd (m+1,n+1) - 1. We show that there are bases to both vector spaces, with entries equal to 0, 1 and -1. When K = Z/(2), the kernel elements of these subspaces are described by rectangular tilings of a special kind. As a corollary, we count the number of tilings of a rectangle of integer sides with a specified set of tiles.

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