scholarly journals Toric Mutations in the dP$_2$ Quiver and Subgraphs of the dP$_2$ Brane Tiling

10.37236/6825 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Yibo Gao ◽  
Zhaoqi Li ◽  
Thuy-Duong Vuong ◽  
Lisa Yang
Keyword(s):  
Planar Graphs ◽  
Perfect Matchings ◽  
Laurent Polynomial ◽  
Theoretical Physics ◽  
Explicit Formulas ◽  
Cluster Variable ◽  
Brane Tiling ◽  
Del Pezzo

Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we study the del Pezzo 2 (dP$_2$) quiver and its associated brane tiling which arise in theoretical physics. Specifically, we prove explicit formulas for all cluster variables generated by toric mutation sequences of the dP$_2$ quiver. Moreover, we associate a subgraph of the dP$_2$ brane tiling to each toric cluster variable such that the sum of weighted perfect matchings of the subgraph equals the Laurent polynomial of the cluster variable.

Spherical functions and local densities on the space of p-adic quaternion Hermitian forms

2021 ◽  
pp. 1-38
Author(s):  
Yumiko Hironaka
Keyword(s):  
Laurent Polynomial ◽  
Spherical Functions ◽  
Plancherel Formula ◽  
Explicit Formulas ◽  
Hermitian Forms ◽  
Spherical Fourier Transform ◽  
Local Densities ◽  
Laurent Polynomial Ring ◽  
Injective Map ◽  
Residual Characteristic

We introduce the space [Formula: see text] of quaternion Hermitian forms of size [Formula: see text] on a [Formula: see text]-adic field with odd residual characteristic, and define typical spherical functions [Formula: see text] on [Formula: see text] and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to [Formula: see text], and define a spherical Fourier transform on the Schwartz space [Formula: see text] which is Hecke algebra [Formula: see text]-injective map into the symmetric Laurent polynomial ring of size [Formula: see text]. Then, we determine the explicit formulas of [Formula: see text] by a method of the author’s former result. In the last section, we give precise generators of [Formula: see text] and determine all the spherical functions for [Formula: see text], and give the Plancherel formula for [Formula: see text].

scholarly journals A Note on Domino Shuffling

10.37236/1056 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
É. Janvresse ◽  
T. de la Rue ◽  
Y. Velenik
Keyword(s):  
Large Class ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Sufficient Condition ◽  
Aztec Diamond

We present a variation of James Propp's generalized domino shuffling, which provides an efficient way to obtain perfect matchings of weighted Aztec diamonds. Our modification is specially tailored to deal with cases when some of the weights are zero. This allows us to tile efficiently a large class of planar graphs, by embedding them in a large enough Aztec diamond. We also give a sufficient condition on the size of the latter diamond for the algorithm to succeed.

scholarly journals The Cube Recurrence

10.37236/1826 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Gabriel D. Carroll ◽  
David Speyer
Keyword(s):  
Recurrence Relation ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Laurent Polynomials ◽  
Combinatorial Interpretation ◽  
Combinatorial Model

We construct a combinatorial model that is described by the cube recurrence, a quadratic recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in ${\Bbb Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky about the structure of these polynomials, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences, including the Somos-6 and Somos-7 sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.

scholarly journals Cluster algebras of unpunctured surfaces and snake graphs

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Gregg Musiker ◽  
Ralf Schiffler
Keyword(s):  
Perfect Matchings ◽  
Laurent Polynomial ◽  
Cluster Algebras ◽  
Polynomial Expansion ◽  
International Audience ◽  
Polynomial Expansions

International audience We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$ . Nous étudions des algèbres amassées avec coefficients principaux associées aux surfaces. Nous présentons une formule directe pour les développements de Laurent des variables amassées dans ces algèbres en terme de couplages parfaits d'un certain graphe $G_{T,\gamma}$ que l'on construit a partir de la surface en recollant des pièces élémentaires que l'on appelle carreaux. Nous donnons aussi une seconde formule pour ces développements en termes de sous-graphes de $G_{T,\gamma}$ .

scholarly journals Annihilating random walks and perfect matchings of planar graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Massimiliano Mattera
Keyword(s):  
Partition Function ◽  
Random Walks ◽  
Mechanical Model ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Statistical Mechanical Model ◽  
International Audience ◽  
Statistical Mechanical ◽  
Annihilating Random Walks

International audience We study annihilating random walks on $\mathbb{Z}$ using techniques of P.W. Kasteleyn and $R$. Kenyonon perfect matchings of planar graphs. We obtain the asymptotic of the density of remaining particles and the partition function of the underlying statistical mechanical model.

scholarly journals The Expected Number of Perfect Matchings in Cubic Planar Graphs

2021 ◽  
pp. 167-174
Author(s):  
Marc Noy ◽  
Clément Requilé ◽  
Juanjo Rué
Keyword(s):  
Planar Graphs ◽  
Perfect Matchings ◽  
Expected Number

scholarly journals Perfect Matchings in Claw-free Cubic Graphs

10.37236/549 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Sang-il Oum
Keyword(s):  
Positive Constant ◽  
Planar Graphs ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Vertex Graph

Lovász and Plummer conjectured that there exists a fixed positive constant $c$ such that every cubic $n$-vertex graph with no cutedge has at least $2^{cn}$ perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic $n$-vertex graph with no cutedge has more than $2^{n/12}$ perfect matchings, thus verifying the conjecture for claw-free graphs.

scholarly journals Upper Bounds for Perfect Matchings in Pfaffian and Planar Graphs

10.37236/2845 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Afshin Behmaram ◽  
Shmuel Friedland
Keyword(s):  
Planar Graphs ◽  
Upper Bounds ◽  
Perfect Matchings ◽  
Pfaffian Graphs ◽  
Better Than ◽  
Fullerene Graphs

We give upper bounds on weighted perfect matchings in Pfaffian graphs. These upper bounds are better than Bregman's upper bounds on the number of perfect matchings. We show that some of our upper bounds are sharp for 3 and 4-regular Pfaffian graphs. We apply our results to fullerene graphs.

scholarly journals Perfect Matchings and Cluster Algebras of Classical Type

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Gregg Musiker
Keyword(s):  
Recent Work ◽  
Cluster Expansion ◽  
Perfect Matchings ◽  
Laurent Polynomial ◽  
Cluster Algebras ◽  
Classical Type ◽  
Combinatorial Interpretation ◽  
Expansion Formula ◽  
Graph Theoretic ◽  
International Audience

International audience In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the $A_n$ case while providing a novel interpretation for the $B_n$, $C_n$, and $D_n$ cases. Dans cet article nous donnons une interprétation combinatoire en termes de théorie des graphes pour les variables de clusters qui apparaissent dans la plupart des algèbres à clusters de type fini. En particulier, nous décrivons une famille de graphes tels qu'une énumération pondérée de leurs matchings parfaits encode le numérateur du polynôme de Laurent associé, tandis que les décompositions du graphe correspondent au dénominateur. Ceci complète les récents travaux de Schiffler et Carroll-Price qui donnent une formule pour le développement d'une variable de cluster dans le cas $A_n$, tout en fournissant une nouvelle interprétation dans les cas $B_n$, $C_n$ et $D_n$.

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