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The Kernel of the Adjacency Matrix of a Rectangular Mesh

Abstract

Given an m × 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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Tomei, Vieira The Kernel of the Adjacency Matrix of a Rectangular Mesh. Discrete Comput Geom 28, 411–425 (2002). https://doi.org/10.1007/s00454-002-2819-z

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