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The Tutte q-Polynomial
[article]

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*. ...

##
###
On the rooted Tutte polynomial

1999
*
Annales de l'Institut Fourier
*

-We are grateful to

doi:10.5802/aif.1709
fatcat:usefmo5j25aqtgavtcl2bnew7q
*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. ...##
###
On the rooted Tutte polynomial
[article]

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). ...

##
###
The multivariate arithmetic Tutte polynomial

2014
*
Transactions of the American Mathematical Society
*

We introduce an arithmetic version of

doi:10.1090/s0002-9947-2014-06092-3
fatcat:oef43cm2tnhtxmllfx3steldlm
*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*. ...##
###
The multivariate arithmetic Tutte polynomial
[article]

2013
*
arXiv
*
pre-print

We introduce an arithmetic version of

arXiv:1207.3629v4
fatcat:4ijlnguknzgr7oa2ir54fh4nrq
*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*. ...##
###
The multivariate arithmetic Tutte polynomial

2012
*
Discrete Mathematics & Theoretical Computer Science
*

International audience We introduce an arithmetic version of

doi:10.46298/dmtcs.3072
fatcat:ayul7aijnzbbtcoc76v4e7ffxm
*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*. ...##
###
Inapproximability of the Tutte polynomial

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] . ...

##
###
Inapproximability of the Tutte polynomial

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] . ...

##
###
Fourientations and the Tutte polynomial

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. ...

##
###
Fourientations and the Tutte Polynomial
[article]

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*. ...

##
###
Some inequalities for the Tutte polynomial
[article]

2010
*
arXiv
*
pre-print

We prove that

arXiv:1004.2639v1
fatcat:niaiwg63xnbhjhve7ty2e4wkwm
*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. ...##
###
The Tutte Polynomial of Complex Reflection Groups
[article]

2020
*
arXiv
*
pre-print

This article computes

arXiv:1911.08792v3
fatcat:kr7s3v5gjvfklizzlulk7erjoq
*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 . ...##
###
Partial Graph Orientations and the Tutte Polynomial
[article]

2015
*
arXiv
*
pre-print

Gessel and Sagan investigated

arXiv:1408.3962v3
fatcat:bage4osqjvdf7chdal6tpnce5i
*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*. ...##
###
On the HOMFLY and Tutte polynomials
[article]

2008
*
arXiv
*
pre-print

We consider

arXiv:0704.0644v2
fatcat:xnzgv3jcwngr7nrnw2as2fyiny
*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*...##
###
A generalization of the Tutte polynomials
[article]

2019
*
arXiv
*
pre-print

In this paper, we introduce

arXiv:1810.04878v2
fatcat:xoi5wel7argqndivtmaqnswjxe
*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 . ...
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