scholarly journals The arithmetic Tutte polynomials of the classical root systems

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Federico Castillo ◽  
Michael Henley
Keyword(s):  
Hyperplane Arrangement ◽  
Tutte Polynomial ◽  
Root Systems ◽  
Field Method ◽  
Topological Invariants ◽  
Signed Graphs ◽  
Tutte Polynomials ◽  
International Audience

International audience Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its <i>arithmetic</i> Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems $A_n, B_n, C_n$, and $D_n$ with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a \emphfinite field method for arithmetic Tutte polynomials, and by enumerating signed graphs with respect to six parameters.

The Tutte polynomial of ideal arrangements

2020 ◽  
Vol 12 (02) ◽  
pp. 2050017
Author(s):  
Hery Randriamaro
Keyword(s):  
Finite Field ◽  
Hyperplane Arrangement ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Root Systems ◽  
Field Method ◽  
Hyperplane Arrangements ◽  
Acyclic Orientations ◽  
The Matrix ◽  
Tutte Polynomials

The Tutte polynomial was originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for certain prime powers of the first variable at the same time. In this paper, we compute the Tutte polynomial of ideal arrangements. These arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of classical root systems, we bring a slight improvement of the finite field method by showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to the studied hyperplane arrangement. Computing the minor set associated to an ideal of classical root systems particularly permits us to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type [Formula: see text], [Formula: see text], and [Formula: see text], we use the formula of Crapo.

scholarly journals Zonotopes, toric arrangements, and generalized Tutte polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Luca Moci
Keyword(s):  
Characteristic Polynomial ◽  
Hilbert Series ◽  
Hyperplane Arrangement ◽  
Tutte Polynomial ◽  
Integral Points ◽  
Poincaré Polynomial ◽  
Tutte Polynomials ◽  
International Audience ◽  
Toric Arrangement ◽  
Positive Coefficients

International audience We introduce a multiplicity Tutte polynomial $M(x,y)$, which generalizes the ordinary one and has applications to zonotopes and toric arrangements. We prove that $M(x,y)$ satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial $M(x,y)$, likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, $M(1,y)$ is the Hilbert series of the related discrete Dahmen-Micchelli space, while $M(x,1)$ computes the volume and the number of integral points of the associated zonotope. On introduit un polynôme de Tutte avec multiplicité $M(x, y)$, qui généralise le polynôme de Tutte ordinaire et a des applications aux zonotopes et aux arrangements toriques. Nous prouvons que $M(x, y)$ satisfait une récurrence de "deletion-restriction'' et a des coefficients positifs. Le polynôme caractéristique et le polynôme de Poincaré d'un arrangement torique sont des spécialisations du polynôme associé $M(x, y)$, de même que les polynômes correspondants pour un arrangement d'hyperplans sont des spécialisations du polynôme de Tutte ordinaire. En outre, $M(1, y)$ est la série de Hilbert de l'espace discret de Dahmen-Micchelli associé, et $M(x, 1)$ calcule le volume et le nombre de points entiers du zonotope associé.

scholarly journals The Tutte polynomial of symmetric hyperplane arrangements

2020 ◽  
Vol 29 (03) ◽  
pp. 2050004
Author(s):  
Hery Randriamaro
Keyword(s):  
Hyperplane Arrangement ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Hyperplane Arrangements ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Tutte Polynomials ◽  
One Year ◽  
The One ◽  
Definition Of

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

scholarly journals A study about the Tutte polynomials of benzenoid chains

2017 ◽  
Vol 5 (1) ◽  
pp. 28-32
Author(s):  
Abdulgani Sahin
Keyword(s):  
Tutte Polynomial ◽  
Signed Graphs ◽  
Tutte Polynomials

Abstract The Tutte polynomials for signed graphs were introduced by Kauffman. In 2012, Fath-Tabar, Gholam-Rezaeı and Ashrafı presented a formula for computing Tutte polynomial of a benzenoid chain. From this point on, we have also calculated the Tutte polynomials of signed graphs of benzenoid chains in this study.

scholarly journals TUTTE POLYNOMIALS OF TENSOR PRODUCTS OF SIGNED GRAPHS AND THEIR APPLICATIONS IN KNOT THEORY

2009 ◽  
Vol 18 (05) ◽  
pp. 561-589 ◽  
Author(s):  
Y. DIAO ◽  
G. HETYEI ◽  
K. HINSON
Keyword(s):  
Tensor Product ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Tensor Products ◽  
Tutte Polynomial ◽  
Signed Graphs ◽  
Substitution Rule ◽  
Simple Substitution ◽  
Jones Polynomials ◽  
Tutte Polynomials

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Our result enables us to compute the Jones polynomials of some large non-alternating knots. The combinatorics used to prove our main result is similar to Tutte's original way of counting "activities" and specializes to a new, perhaps simpler proof of the known formulas for the ordinary Tutte polynomial of the tensor product of unsigned graphs or matroids.

scholarly journals Cumulants of the q-semicircular law, Tutte polynomials, and heaps

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Matthieu Josuat-Vergès
Keyword(s):  
Tutte Polynomial ◽  
Combinatorial Interpretation ◽  
Poisson Law ◽  
Semicircular Law ◽  
Combinatorial Model ◽  
Tutte Polynomials ◽  
International Audience ◽  
Nonnegative Coefficients

International audience The q-semicircular law as introduced by Bożejko and Speicher interpolates between the Gaussian law and the semicircular law, and its moments have a combinatorial interpretation in terms of matchings and crossings. We prove that the cumulants of this law are, up to some factor, polynomials in q with nonnegative coefficients. This is done by showing that they are obtained by an enumeration of connected matchings, weighted by the evaluation at (1,q) of a Tutte polynomial. The two particular cases q=0 and q=2 have also alternative proofs, related with the fact that these particular evaluation of the Tutte polynomials count some orientations on graphs. Our methods also give a combinatorial model for the cumulants of the free Poisson law. La loi q-semicirculaire introduite par Bożejko et Speicher interpole entre la loi gaussienne et la loi semi-circulaire, et ses moments ont une interprétation combinatoire en termes de couplages et croisements. Nous prouvons que les cumulants de cette loi sont, à un facteur près, des polynômes en q à coefficients positifs. La méthode consiste à obtenir ces cumulants par une énumération de couplages connexes, pondérés par l’évaluation en (1,q) d'un polynôme de Tutte. Les cas particuliers q=0 et q=2 ont une preuve alternative, reliè au fait que des évaluations particulières du polynôme de Tutte comptent certaines orientations de graphes. Nos méthodes donnent aussi un modèle combinatoire aux cumulants de la loi de Poisson libre.

scholarly journals Arithmetic matroids and Tutte polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Michele D'Adderio ◽  
Luca Moci
Keyword(s):  
Abelian Group ◽  
Tutte Polynomial ◽  
Finitely Generated ◽  
Combinatorial Interpretation ◽  
Tutte Polynomials ◽  
International Audience

International audience We introduce the notion of arithmetic matroid, whose main example is provided by a list of elements in a finitely generated abelian group. We study the representability of its dual, and, guided by the geometry of toric arrangements, we give a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula. Nous introduisons la notion de matroï de arithmètique, dont le principal exemple est donnè par une liste d'élèments dans un groupe abèlien fini. Nous ètudions la reprèsentabilitè de son dual, et, guidè par la gèomètrie des arrangements toriques, nous donnons une interprètation combinatoire du polynôme de Tutte arithmètique associèe, ce qui peut être vu comme une gènèralisation de la formule de Crapo.

scholarly journals Computing Tutte Polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Michael Monagan
Keyword(s):  
Recent Result ◽  
Ad Hoc ◽  
Tutte Polynomial ◽  
Nous Pouvons ◽  
Sparse Graphs ◽  
Vertex Ordering ◽  
Tutte Polynomials ◽  
International Audience

International audience We present a new edge selection heuristic and vertex ordering heuristic that together enable one to compute the Tutte polynomial of much larger sparse graphs than was previously doable. As a specific example, we are able to compute the Tutte polynomial of the truncated icosahedron graph using our Maple implementation in under 4 minutes on a single CPU. This compares with a recent result of Haggard, Pearce and Royle whose special purpose C++ software took one week on 150 computers. Nous présentons deux nouvelles heuristiques pour le calcul du polynôme de Tutte de graphes de faible densité, basées sur les principes de sélection d'arêtes et d'arrangement linéaire de sommets, et qui permettent de traiter des graphes de bien plus grande tailles que les méthodes existantes. Par exemple, en utilisant notre implémentation en Maple, nous pouvons calculer le polynôme de Tutte de l'isocahédron tronqué en moins de 4 minutes sur un ordinateur à processeur unique, alors qu'un programme ad-hoc récent de Haggard, Pearce et Royle, utilisant 150 ordinateurs, a nécessité une semaine de calcul pour obtenir le même résultat.

scholarly journals Dichromatic polynomial for graph of a (2,n)-torus knot

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdulgani Şahin
Keyword(s):  
Tutte Polynomial ◽  
Signed Graph ◽  
Torus Knots ◽  
Torus Knot ◽  
Tutte Polynomials ◽  
The Relationship ◽  
Dichromatic Polynomial

AbstractIn this study, we introduce the relationship between the Tutte polynomials and dichromatic polynomials of (2,n)-torus knots. For this aim, firstly we obtain the signed graph of a (2,n)-torus knot, marked with {+} signs, via the regular diagram of its. Whereupon, we compute the Tutte polynomial for this graph and find a generalization through these calculations. Finally we obtain dichromatic polynomial lying under the unmarked states of the signed graph of the (2,n)-torus knots by the generalization.

scholarly journals Graph polynomials and symmetries

2019 ◽  
Vol 18 (09) ◽  
pp. 1950172 ◽  
Author(s):  
Nafaa Chbili
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Prime Order ◽  
Tutte Polynomial ◽  
Quotient Graph ◽  
Graph Polynomials ◽  
Tutte Polynomials

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

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