Domino Tilings and Determinants
Consider an arbitrary simply connected figure F on the square grid and its dual graph (vertices correspond to cells, edges correspond to cells sharing a common side). We investigate the relationship between the determinant of the adjacency matrix of the graph and the domino tilings of the figure F. We prove that in the case where all the tilings can be split into pairs such that the numbers of vertical dominoes in each pair differ by one, then det A F = 0. And in the case where all the tilings except one can be split into such pairs, det A F = (−1) s , where s is half the area of the figure F. Bibliography: 6 titles.
KeywordsAdjacency Matrix Integer Point Dual Graph Black Vertex White Vertex
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