The Kernel of the Adjacency Matrix of a Rectangular Mesh
The Kernel of the Adjacency Matrix of a Rectangular Mesh
复制标题
矩形网格邻接矩阵的核
DOI：
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发表时间：
2002
影响因子：
0.8
通讯作者：
Tania Vieira
中科院分区：
文献类型：
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作者：
C. Tomei;Tania Vieira
给定一个m × n的矩形网格，它的邻接矩阵A只有整数项，可以解释为任意域K上向量空间之间的映射。我们描述了A的核：它是两个自然子空间的直接和，它们的维数为$lceil c/2 ceil$和$lfloor c/2 floor$，其中c = gcd (m+1,n+1) - 1。我们证明了两个向量空间都有基底，它们的元素分别为0,1和-1。当K = Z/(2)时，这些子空间的核元素用一种特殊的矩形平铺来描述。作为推论，我们用一组指定的瓷砖来计算一个矩形的整数边的瓷砖数量。
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 $lceil c/2
ceil$ and $lfloor c/2
floor$ , 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.