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The kernel of the adjacency matrix of a rectangular mesh
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 and , 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.Comment: 15 pages, 17 figure