Determinants and perfect matchings
نویسندگان
چکیده
منابع مشابه
Determinants and perfect matchings
We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula for the determinant is replaced by the number of crossings in the Brauer diagram. This interpretation naturally explains why the determinant of an even antisy...
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We give a characterization of the effect of sequences of pivot operations on a graph by relating it to determinants of adjacency matrices. This allows us to deduce that two sequences of pivot operations are equivalent iff they contain the same set S of vertices (modulo two). Moreover, given a set of vertices S, we characterize whether or not such a sequence using precisely the vertices of S exi...
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In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers (N. Elkies et al., J. Algebraic Combin. 1 (1992), 111– 132; B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, Cambridge, MA, 1991; J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this pa...
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In 1961, P.W. Kasteleyn enumerated the domino tilings of a 2n × 2n chessboard. His answer was always a square or double a square (we call such a number "squarish"), but he did not provide a combinatorial explanation for this. In the present thesis, we prove by mostly combinatorial arguments that the number of matchings of a large class of graphs with 4-fold rotational symmetry is squarish; our ...
متن کاملPerfect Matchings in Edge-Transitive Graphs
We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2013
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2012.08.007