Tutte Polynomial Research Articles (Page 1)

Weak maps and the Tutte polynomial

Abstract The Tutte polynomial is a fundamental invariant of matroids. The polymatroid Tutte polynomial $\mathscr{T}_{P}(x,y)$, introduced by Bernardi, Kálmán, and Postnikov, is an extension of the classical Tutte polynomial from matroids to polymatroids $P$. In this paper, we first obtain a deletion-contraction formula for $\mathscr{T}_{P}(x,y)$. Then we prove two natural properties of coefficientwise monotonicity, one for containment and one for minors, both for the interior polynomial $x^{n}\mathscr{T}_{P}(x^{-1},1)$ and the exterior polynomial $y^{n}\mathscr{T}_{P}(1,y^{-1})$, where $P$ is a polymatroid over $[n]$. We show by an example that these monotonicity properties do not extend to $\mathscr{T}_{P}(x,y)$. Using deletion-contraction, we obtain formulas for the coefficients of terms of degree $n-1$ in $\mathscr{T}_{P}(x,y)$. Finally, we characterize hypergraphs $\mathcal{H}=(V,E)$ such that the coefficient of $y^{k}$ in the exterior polynomial of the associated polymatroid $P_{\mathcal{H}}$ attains its maximal value $\binom{|V|+k-2}{k}$ for all $k$ up to some bound.

Corrigendum to “On maximum graphs in Tutte polynomial posets” [Discrete Appl. Math. 339 (2023) 78–88

Tutte polynomials of matroids as universal valuative invariants

We consider the Tutte polynomial of three classes of greedoids: those arising from rooted graphs, rooted digraphs and binary matrices. We establish the computational complexity of evaluating each of these polynomials at each fixed rational point $(x,y)$. In each case we show that evaluation is $\#$P-hard except for a small number of exceptional cases when there is a polynomial time algorithm. In the binary case, establishing $\#$P-hardness along one line relies on Vertigan's unpublished result on the complexity of counting bases of a binary matroid. For completeness, we include an appendix providing a proof of this result.

Perturbative calculations involving fermion loops in quantum field theories require tracing over Dirac matrices. A simple way to regulate the divergences that generically appear in these calculations is dimensional regularisation, which has the consequence of replacing 4-dimensional Dirac matrices with d-dimensional counterparts for arbitrary complex values of d. In this work, a connection between traces of d-dimensional Dirac matrices and computations of the Tutte polynomial of associated graphs is proven. The time complexity of computing Dirac traces is analysed by this connection, and improvements to algorithms for computing Dirac traces are proposed.

Inconsequential results on the Merino-Welsh conjecture for Tutte polynomials

Coefficients of the Tutte polynomial and minimal edge cuts of a graph

Abstract Dominic Welsh was born in Port Talbot on 29 August 1938, the eldest child in a family of educators, and died in Oxford on 30 November 2023. He was the first student from his school to attend the University of Oxford, where he remained for the rest of his life as a Fellow of Merton College and a Professor of the University. He combined excellence as tutor and supervisor over nearly 40 years with a distinguished research record in probability and discrete mathematics, where he excelled in both original and expository work. With his DPhil supervisor John Hammersley, he introduced first‐passage percolation, and in so doing formulated and proved the first subadditive ergodic theorem. His is the ‘W’ in the ‘RSW’ method that is now central to the theory of random planar media. He was a pioneer in matroid theory with numerous significant results and conjectures, and his monograph has been influential. He worked on computational complexity and particularly the complexity of computing the Tutte polynomial. Throughout his career, he inspired generations of undergraduates and postgraduates, and through his personal enthusiasm and warmth he helped develop a community of scholars in aspects of combinatorics who remember him with love and respect.

A direct proof of well-definedness for the polymatroid Tutte polynomial

The beta invariant is closely related to the Chromatic and Tutte Polynomials and has been extensively studied, see Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22], Crapo [A higher invariant for matroids, J. Combin. Theory 2 (1967) 406–417], Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11], Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277], and others. Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11] established that a 3-connected graph with [Formula: see text] vertices possesses a beta invariant of at least [Formula: see text], reaching equality only when the graph is a wheel or the Prism. Additionally, Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277] provided characterizations for matroids with beta invariant values of two, three, and four. This paper extends the findings of Lee and Wu by offering a comprehensive characterization of all 3-connected graphs with [Formula: see text] vertices that have a beta invariant of either [Formula: see text] or [Formula: see text]. As a consequence, we prove that any 3-connected graph with at least [Formula: see text] vertices other than a Wheel has a beta invariant of at least [Formula: see text].

We make a systematic study of matroidal mixed Eulerian numbers which are certain intersection numbers in the matroid Chow ring generalizing the mixed Eulerian numbers introduced by Postnikov. These numbers are shown to be valuative and obey a log-concavity relation. We establish recursion formulas and use them to relate matroidal mixed Eulerian numbers to the characteristic and Tutte polynomials, reproving results of Huh–Katz and Berget–Spink–Tseng. Generalizing Postnikov, we show that these numbers are equal to certain weighted counts of binary trees. Lastly, we study these numbers for perfect matroid designs, proving that they generalize the remixed Eulerian numbers of Nadeau–Tewari.

Delta-matroids are “type B” generalizations of matroids in the same way that maximal orthogonal Grassmannians are generalizations of Grassmannians. A delta-matroid analogue of the Tutte polynomial of a matroid is the interlace polynomial. We give a geometric interpretation for the interlace polynomial via the K K -theory of maximal orthogonal Grassmannians. To do so, we develop a new Hirzebruch–Riemann–Roch-type formula for the type B permutohedral variety.

Abstract We study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a stressed subset. This framework provides a new combinatorial characterization of the class of (elementary) split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which, in turn, can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on these matroids depend solely on the behavior of the invariant on a tractable subclass of Schubert matroids. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan–Lusztig polynomials, the Whitney numbers of the first and second kinds, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's ‐polynomials, as well as Chow rings of matroids and their Hilbert–Poincaré series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.

Kramers–Wannier duality and Tutte polynomials

Tutte polynomial for a small world connected copies of Farey graphs

Abstract We initiate the axiomatic study of affine oriented matroids (AOMs) on arbitrary ground sets, obtaining fundamental notions such as minors, reorientations and a natural embedding into the frame work of Complexes of Oriented Matroids. The restriction to the finitary case (FAOMs) allows us to study tope graphs and covector posets, as well as to view FAOMs as oriented finitary semimatroids. We show shellability of FAOMs and single out the FAOMs that are affinely homeomorphic to $$\mathbb {R}^n$$ R n . Finally, we study group actions on AOMs, whose quotients in the case of FAOMs are a stepping stone towards a general theory of affine and toric pseudoarrangements. Our results include applications of the multiplicity Tutte polynomial of group actions of semimatroids, generalizing enumerative properties of toric arrangements to a combinatorially defined class of arrangements of submanifolds. This answers partially a question by Ehrenborg and Readdy.

The Merino–Welsh conjecture is false for matroids

Convolution formulas for multivariate arithmetic Tutte polynomials

We show that for any non-real algebraic number q, such that|q-1|>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|q-1|>1$$\\end{document} or ℜ(q)>32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re(q)>\\frac{3}{2}$$\\end{document} it is #P-hard to computea multiplicative (resp. additive) approximation to the absolutevalue (resp. argument) of the chromatic polynomial evaluated at q on planar graphs. This implies #P-hardness for allnon-real algebraic q on the family of all graphs. We, moreover,prove several hardness results for q, such that |q-1|≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|q-1|\\leq 1$$\\end{document}.Our hardness results are obtained by showing that a polynomial timealgorithm for approximately computing the chromaticpolynomial of a planar graph at non-real algebraic q (satisfyingsome properties) leads to a polynomial time algorithm forexactly computing it, which is known to be hard by a resultof Vertigan. Many of our results extend in fact to the more generalpartition function of the random cluster model, a well-knownreparametrization of the Tutte polynomial.