Graph Polynomials and Their Applications I: The Tutte Polynomial

Graph polynomials and symmetries

Journal of Algebra and Its Applications ◽

10.1142/s021949881950172x ◽

2019 ◽

Vol 18 (09) ◽

pp. 1950172 ◽

Cited By ~ 1

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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Graphs, ribbon graphs, and polynomials

Combinatorial Physics ◽

10.1093/oso/9780192895493.003.0002 ◽

2021 ◽

pp. 7-16

Author(s):

Adrian Tanasa

Keyword(s):

Graph Theory ◽

Tutte Polynomial ◽

Graph Polynomials ◽

Ribbon Graphs

In this chapter we present some notions of graph theory that will be useful in the rest of the book. It is worth emphasizing that graph theorists and theoretical physicists adopt, unfortunately, different terminologies. We present here both terminologies, such that a sort of dictionary between these two communities can be established. We then extend the notion of graph to that of maps (or of ribbon graphs). Moreover, graph polynomials encoding these structures (the Tutte polynomial for graphs and the Bollobás–Riordan polynomial for ribbon graphs) are presented.

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Graph polynomials and link invariants as positive type functions on Thompson’s group F

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500068 ◽

2019 ◽

Vol 28 (02) ◽

pp. 1950006 ◽

Cited By ~ 2

Author(s):

Valeriano Aiello ◽

Roberto Conti

Keyword(s):

Elementary Proof ◽

Tutte Polynomial ◽

Positive Type ◽

Polynomial Invariants ◽

Graph Polynomials ◽

Link Invariants ◽

Kauffman Bracket ◽

Model Interpretation ◽

Mechanics Model ◽

Thompson’S Group

In a recent paper, Jones introduced a correspondence between elements of the Thompson group [Formula: see text] and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of [Formula: see text]. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of [Formula: see text]-colorings and the Tutte polynomial, can be viewed as positive definite functions on [Formula: see text].

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The Equivalence of Two Graph Polynomials and a Symmetric Function

Combinatorics Probability Computing ◽

10.1017/s0963548309009845 ◽

2009 ◽

Vol 18 (4) ◽

pp. 601-615 ◽

Cited By ~ 10

Author(s):

CRIEL MERINO ◽

STEVEN D. NOBLE

Keyword(s):

Symmetric Function ◽

Tutte Polynomial ◽

Graph Polynomials ◽

Definition Of ◽

Natural Way

The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial, due to Stanley, are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to strengthen them, which also captures Tutte's universal V-function as a specialization. We show that the equivalence remains true for the strong functions, thus answering a question raised by Dominic Welsh.

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A NOTE ON THE TUTTE POLYNOMIAL AND THE AUTOMORPHISM GROUP OF A GRAPH

Asian-European Journal of Mathematics ◽

10.1142/s1793557114500016 ◽

2014 ◽

Vol 07 (01) ◽

pp. 1450001 ◽

Cited By ~ 2

Author(s):

Nafaa Chbili

Keyword(s):

Automorphism Group ◽

Short Note ◽

Tutte Polynomial ◽

Necessary Condition ◽

Graph Polynomials

A graph G is said to be p-periodic if the automorphism group Aut(G) contains an element of order p which preserves no edges. In this short note, we investigate the behavior of graph polynomials (Negami and Tutte) with respect to graph periodicity. In particular, we prove that if p is a prime, then the coefficients of the Tutte polynomial of such a graph satisfy a certain necessary condition.

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Types of embedded graphs and their Tutte polynomials

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004119000161 ◽

2019 ◽

Vol 169 (2) ◽

pp. 255-297 ◽

Cited By ~ 1

Author(s):

STEPHEN HUGGETT ◽

IAIN MOFFATT

Keyword(s):

Systematic Approach ◽

Tutte Polynomial ◽

Embedded Graphs ◽

Graph Polynomials ◽

Ribbon Graphs ◽

Duality Relations ◽

Graphs On Surfaces ◽

Tutte Polynomials

AbstractWe take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a ‘Tutte polynomial’) for each class. We relate these to graph polynomials in the literature, including the Bollobás–Riordan, Krushkal and Las Vergnas polynomials, and give state-sum formulations, duality relations, deleton-contraction relations, and quasi-tree expansions for each of them.

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The Tutte polynomial of symmetric hyperplane arrangements

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500042 ◽

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.

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