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The Tutte q-Polynomial [article]

Guus Bollen, Henry Crapo, Relinde Jurrius
2017 arXiv   pre-print
The q-analogue of the passage from a Tutte polynomial to its corresponding RGF is straight-forward, but the analogue of the reverse process x → (x-1), y → (y-1) is more delicate.  ...  Tutte polynomials τ(x,y) of matroids are calculated either by recursion over deletion/contraction of single elements, by an enumeration of bases with respect to internal/external activities, or by substitution  ...  Tutte polynomial.  ... 
arXiv:1707.03459v1 fatcat:x2lh4x6xmjcmxmsocibwcsmee4

On the rooted Tutte polynomial

F. Y. Wu, C. King, W. T. Lu
1999 Annales de l'Institut Fourier  
-We are grateful to the referee for providing an independent proof of Proposition 1 and numerous suggested improvements on an earlier version of this paper.  ...  -For n == 1, the duality relation (23) for the rooted Tutte polynomial becomes the duality relation (5) for the Tutte polynomial.  ...  The rooted Tutte polynomial. We extend the definition (4) to a rooted Tutte polynomial. A vertex is rooted, or is a root, if it is colored with a prescribed (fixed) color.  ... 
doi:10.5802/aif.1709 fatcat:usefmo5j25aqtgavtcl2bnew7q

On the rooted Tutte polynomial [article]

F. Y. Wu, C. King, W. T. Lu
1998 arXiv   pre-print
The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings.  ...  We establish a number of results pertaining to the rooted Tutte polynomial, including a duality relation in the case that all roots reside around a single face of a planar graph.  ...  This work is supported in part by the National Science Foundation grants DMR-9614170 (FYW and WTL) and DMS-9705779 (CK).  ... 
arXiv:cond-mat/9812202v1 fatcat:yruityhjxfcg3iwnbx3kvomhcy

The multivariate arithmetic Tutte polynomial

Petter Brändén, Luca Moci
2014 Transactions of the American Mathematical Society  
We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two.  ...  We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids.  ...  can be obtained by specializing the Tutte polynomial.  ... 
doi:10.1090/s0002-9947-2014-06092-3 fatcat:oef43cm2tnhtxmllfx3steldlm

The multivariate arithmetic Tutte polynomial [article]

Petter Brändén, Luca Moci
2013 arXiv   pre-print
We introduce an arithmetic version of the multivariate Tutte polynomial, and (for representable arithmetic matroids) a quasi-polynomial that interpolates between the two.  ...  We give a new and more general proof of the positivity of the coefficients of the arithmetic Tutte polynomial, and (in the representable case) a geometrical interpretation of them.  ...  can be obtained by specializing the Tutte polynomial.  ... 
arXiv:1207.3629v4 fatcat:4ijlnguknzgr7oa2ir54fh4nrq

The multivariate arithmetic Tutte polynomial

Petter Brändèn, Luca Moci
2012 Discrete Mathematics & Theoretical Computer Science  
International audience We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two.  ...  We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids.  ...  can be obtained by specializing the Tutte polynomial.  ... 
doi:10.46298/dmtcs.3072 fatcat:ayul7aijnzbbtcoc76v4e7ffxm

Inapproximability of the Tutte polynomial

Leslie Ann Goldberg, Mark Jerrum
2008 Information and Computation  
The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph.  ...  Jaeger, Vertigan and Welsh have completely mapped the complexity of exactly computing the Tutte polynomial.  ...  The Tutte polynomial is sometimes referred to as the "Whitney-Tutte" polynomial, or the "dichromatic polynomial". See [20, 22] .  ... 
doi:10.1016/j.ic.2008.04.003 fatcat:apyqrues7zfldleoodpbrjjdtm

Inapproximability of the Tutte polynomial

Leslie Ann Goldberg, Mark Jerrum
2007 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07  
The Tutte polynomial of a graph G is a two-variable polynomial T(G; x,y) that encodes many interesting properties of the graph.  ...  Jaeger et al. have completely mapped the complexity of exactly computing the Tutte polynomial.  ...  The Tutte polynomial is sometimes referred to as the "Whitney-Tutte" polynomial, or the "dichromatic polynomial". See [20, 22] .  ... 
doi:10.1145/1250790.1250858 dblp:conf/stoc/GoldbergJ07 fatcat:nma77trm6fhwpn5w2xkwd42gka

Fourientations and the Tutte polynomial

Spencer Backman, Sam Hopkins
2017 Research in the Mathematical Sciences  
Tutte polynomial evaluations of the form We introduce an intersection lattice of 64 cut-cycle fourientation classes enumerated by generalized Tutte polynomial evaluations of this form.  ...  NATO Advanced Study Institute series, series C: mathematical and physical sciencesLas Vergnas (Tutte polynomial of a morphism of matroids 6.  ...  Acknowledgements We thank Olivier Bernardi for some enlightening discussions about the Tutte polynomial and for his encouragement with our investigation of fourientations.  ... 
doi:10.1186/s40687-017-0107-z fatcat:kukntojwjzhtbpg3pozjior3x4

Fourientations and the Tutte Polynomial [article]

Spencer Backman, Sam Hopkins
2015 arXiv   pre-print
Tutte polynomial evaluations of the form (k+m)^n-1(k+l)^gT(α k + β l + m/k+m,γ k + l + δ m/k+l) for α,γ∈{0,1,2} and β, δ∈{0,1}.  ...  We introduce an intersection lattice of 64 cut-cycle fourientation classes enumerated by generalized Tutte polynomial evaluations of this form.  ...  the Tutte polynomial.  ... 
arXiv:1503.05885v3 fatcat:5b6phurqnbd3ngabpmgus4mkdq

Some inequalities for the Tutte polynomial [article]

L. E. Chavez-Lomelí, C. Merino, S. D. Noble, M. Ramírez-Ibañez
2010 arXiv   pre-print
We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant.  ...  Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*.  ...  The Tutte polynomials of the graphs at the top of Fig. 1 are convex functions while the Tutte polynomial of the graph at the bottom is neither convex nor concave.  ... 
arXiv:1004.2639v1 fatcat:niaiwg63xnbhjhve7ty2e4wkwm

The Tutte Polynomial of Complex Reflection Groups [article]

Hery Randriamaro
2020 arXiv   pre-print
This article computes the Tutte polynomial of the hyperplane arrangements associated to the complex reflection groups.  ...  The calculations are based on both formulas of De Concini and Procesi for Tutte polynomial and the normaliser of parabolic subgroups in complex reflection groups determined by Krishnasamy and Taylor.  ...  The Tutte polynomial T G (x, y) associated to G is the Tutte polynomial of A G , that is T G (x, y) := B⊆A G (x − 1) rk A G −rk B (y − 1) #B−rk B .  ... 
arXiv:1911.08792v3 fatcat:kr7s3v5gjvfklizzlulk7erjoq

Partial Graph Orientations and the Tutte Polynomial [article]

Spencer Backman
2015 arXiv   pre-print
Gessel and Sagan investigated the Tutte polynomial, T(x,y) using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is 2^gT(3,1/2).  ...  We conclude with edge chromatic generalizations of the quantities presented, which allow for a new interpretation of the reliability polynomial for all probabilities, p with 0 < p <1/2.  ...  Additional thanks to Sam for suggesting that I investigate the relationship between my work and the reliability polynomial.  ... 
arXiv:1408.3962v3 fatcat:bage4osqjvdf7chdal6tpnce5i

On the HOMFLY and Tutte polynomials [article]

Iain Moffatt
2008 arXiv   pre-print
We consider the question 'to what extent does the Tutte polynomial determine the HOMFLY polynomial of any knot?'  ...  We show that the HOMFLY polynomial of a knot is determined by Tutte polynomials of plane graphs associated to the knot.  ...  Given this realization of the Tutte polynomial as the HOMFLY polynomial of a class of links (note that this is a proper subset of the set of links), it is natural to ask to what extent the Tutte polynomial  ... 
arXiv:0704.0644v2 fatcat:xnzgv3jcwngr7nrnw2as2fyiny

A generalization of the Tutte polynomials [article]

Tsuyoshi Miezaki, Manabu Oura, Tadashi Sakuma, Hidehiro Shinohara
2019 arXiv   pre-print
In this paper, we introduce the concept of the Tutte polynomials of genus g and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials.  ...  The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus g are also matroid invariants.  ...  Tutte polynomials of genus g We now present the concept of the Tutte polynomial of genus g. For every element λ ∈ Λ 2 , and A i ⊂ E, let us denote A ∩(λ) := ∩ i∈λ A i and A ∪(λ) := ∪ i∈λ A i .  ... 
arXiv:1810.04878v2 fatcat:xoi5wel7argqndivtmaqnswjxe
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