Types of embedded graphs and their Tutte polynomials

2019 ◽  
Vol 169 (2) ◽  
pp. 255-297 ◽  
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.

scholarly journals Graphs, Links, and Duality on Surfaces

2010 ◽  
Vol 20 (2) ◽  
pp. 267-287 ◽  
Author(s):  
VYACHESLAV KRUSHKAL
Keyword(s):  
Planar Graphs ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Natural Duality ◽  
Polynomial Invariant ◽  
Topological Duality ◽  
Ribbon Graphs ◽  
Virtual Links ◽  
Graphs On Surfaces ◽  
Tutte Polynomials

We introduce a polynomial invariant of graphs on surfaces,PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result forPG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs,PGspecializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomialPG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

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.

Graphs, ribbon graphs, and polynomials

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.

scholarly journals Evaluations of Topological Tutte Polynomials

2014 ◽  
Vol 24 (3) ◽  
pp. 556-583 ◽  
Author(s):  
J. ELLIS-MONAGHAN ◽  
I. MOFFATT
Keyword(s):  
Tensor Product ◽  
General Framework ◽  
Tutte Polynomial ◽  
Ribbon Graph ◽  
Reduction Formula ◽  
Embedded Graphs ◽  
Chromatic Polynomials ◽  
Topological Transition ◽  
Combinatorial Interpretations ◽  
Tutte Polynomials

We find new properties of the topological transition polynomial of embedded graphs, Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, R(G), and the topological Penrose polynomial, P(G). The general framework provided by Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G), R(G), and the Tutte polynomial, T(G), as sums of chromatic polynomials of graphs derived from G, show that these polynomials count k-valuations of medial graphs, show that R(G) counts edge 3-colourings, and reformulate the Four Colour Theorem in terms of R(G). We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of P(G) and R(G).

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 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 The Arithmetic Tutte polynomial of two matrices associated to Trees

2018 ◽  
Vol 6 (1) ◽  
pp. 310-322
Author(s):  
R. B. Bapat ◽  
Sivaramakrishnan Sivasubramanian
Keyword(s):  
Analogous Result ◽  
Polynomial Matrix ◽  
Tutte Polynomial ◽  
Distance Matrix ◽  
Minimum Volume ◽  
Maximum Volume ◽  
Recent Interest ◽  
Minimum Volume Ellipsoid ◽  
Tutte Polynomials ◽  
Exponential Distance

Abstract Arithmetic matroids arising from a list A of integral vectors in Zn are of recent interest and the arithmetic Tutte polynomial MA(x, y) of A is a fundamental invariant with deep connections to several areas. In this work, we consider two lists of vectors coming from the rows of matrices associated to a tree T. Let T = (V, E) be a tree with |V| = n and let LT be the q-analogue of its Laplacian L in the variable q. Assign q = r for r ∈ ℤ with r/= 0, ±1 and treat the n rows of LT after this assignment as a list containing elements of ℤn. We give a formula for the arithmetic Tutte polynomial MLT (x, y) of this list and show that it depends only on n, r and is independent of the structure of T. An analogous result holds for another polynomial matrix associated to T: EDT, the n × n exponential distance matrix of T. More generally, we give formulae for the multivariate arithmetic Tutte polynomials associated to the list of row vectors of these two matriceswhich shows that even the multivariate arithmetic Tutte polynomial is independent of the tree T. As a corollary, we get the Ehrhart polynomials of the following zonotopes: - ZEDT obtained from the rows of EDT and - ZLT obtained from the rows of LT. Further, we explicitly find the maximum volume ellipsoid contained in the zonotopes ZEDT, ZLT and show that the volume of these ellipsoids are again tree independent for fixed n, q. A similar result holds for the minimum volume ellipsoid containing these zonotopes.

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

2019 ◽  
Vol 28 (02) ◽  
pp. 1950006 ◽  
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].

On the computational complexity of the Jones and Tutte polynomials

1990 ◽  
Vol 108 (1) ◽  
pp. 35-53 ◽  
Author(s):  
F. Jaeger ◽  
D. L. Vertigan ◽  
D. J. A. Welsh
Keyword(s):  
Computational Complexity ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Wide Range ◽  
Tutte Polynomials ◽  
Special Points ◽  
Alternating Link ◽  
Special Case

AbstractWe show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, y)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y)-plane is #P-hard except when (x − 1)(y − 1) = 1 or when (x, y) equals (1, 1), (−1, −1), (0, −1), (−1, 0), (i, −i), (−i, i), (j, j2), (j2, j) where j = e2πi/3

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