# Trees and Matchings

@article{Kenyon2000TreesAM, title={Trees and Matchings}, author={Richard W. Kenyon and James Gary Propp and David Bruce Wilson}, journal={Electron. J. Comb.}, year={2000}, volume={7} }

In this article, Temperley's bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees (often called arborescences) of any planar graph $G$ can be put into a one-to-one weight… Expand

#### 116 Citations

Combinatorics of Tripartite Boundary Connections for Trees and Dimers

- Computer Science, Mathematics
- Electron. J. Comb.
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These Pfaffian formulas are used to give exact expressions for reconstruction: reconstructing the conductances of a planar graph from boundary measurements, and similar theorems for the double-dimer model on bipartite planar graphs are proved. Expand

Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole

- Mathematics
- 2007

Abstract
We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid Gn of order n is similar to the disjoint union of two copies of the… Expand

Enumerating spanning trees of graphs with an involution

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2009

It is shown that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of the sums of weighted trees of two weighted graphs with a smaller size determined by the involution of G. Expand

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- Mathematics, Physics
- 2000

The problem of enumerating spanning trees on graphs and lattices is considered. We obtain bounds on the number of spanning trees NST and establish inequalities relating the numbers of spanning trees… Expand

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

- Mathematics, Computer Science
- Theory of Computing Systems
- 2009

A deterministic Logspace procedure, which, given a bipartite planar graph on n vertices, assigns O(log n) bits long weights to its edges so that the minimum weight perfect matching in the graph becomes unique, and tries to find the lower bound on the number of bits needed for deterministically isolating a perfect matching. Expand

On the Number of α-Orientations

- 2007

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice… Expand

Ja n 20 07 On the Number of α-Orientations

- 2007

We deal with the enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings and ice models. The… Expand

A Bijection Theorem for Domino Tilings with Diagonal Impurities

- Mathematics
- 2010

We consider the dimer problem on a planar non-bipartite graph G, where there are two types of dimers one of which we regard as impurities. Computer simulations reveal a reminiscence of the Cheerios… Expand

Kasteleyn cokernels and perfect matchings on planar bipartite graphs

- Mathematics
- 2018

The determinant method of Kasteleyn gives a method of computing the number of perfect matchings of a planar bipartite graph. In addition, results of Bernardi exhibit a bijection between spanning… Expand

Spanning forests and the vector bundle Laplacian

- Mathematics, Physics
- 2011

The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and… Expand

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