Multivariate Tutte Polynomial Research Articles

Convolution formulas for multivariate arithmetic Tutte polynomials

The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. Applying our results, several known convolution formulas including [5, Theorem 10.9 and Corollary 10.10] and [1, Theorems 1 and 4] are proved by a purely combinatorial proof. The proofs presented here are significantly shorter than the previous ones. In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid.

Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation

We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle ($Y-\Delta$) transformation at the critical point $n=2$. We deduce the theorem of Matiyasevich and its inverse from the Biggs formula, and we apply this relation to construct the recursion on the parameter $n$. We provide two different proofs of the Zamolodchikov tetrahedron equation satisfied by the star-triangle transformation in the case of $n=2$ multivariate Tutte polynomial, we extend the latter to the case of valency 2 points and show that the Biggs formula and the star-triangle transformation commute.

Lorentzian polynomials

We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive eigenvalue at any point on the positive orthant. This property can be seen as an analog of the Hodge--Riemann relations for Lorentzian polynomials. Lorentzian polynomials are intimately connected to matroid theory and negative dependence properties. We show that matroids, and more generally M-convex sets, are characterized by the Lorentzian property, and develop a theory around Lorentzian polynomials. In particular, we provide a large class of linear operators that preserve the Lorentzian property and prove that Lorentzian measures enjoy several negative dependence properties. We also prove that the class of tropicalized Lorentzian polynomials coincides with the class of M-convex functions in the sense of discrete convex analysis. The tropical connection is used to produce Lorentzian polynomials from M-convex functions. We give two applications of the general theory. First, we prove thatthe homogenized multivariate Tutte polynomial of a matroid is Lorentzian whenever the parameter $q$ satisfies $0 \lt q \le 1$. Consequences are proofs of the strongest Mason's conjecture from 1972 and negative dependence properties of the random cluster model in statistical physics. Second, we prove that the multivariate characteristic polynomial of an M-matrix is Lorentzian. This refines a result of Holtz who proved that the coefficients of the characteristic polynomial of an M-matrix form an ultra log-concave sequence.

Logarithmic concavity for morphisms of matroids

Morphisms of matroids are combinatorial abstractions of linear maps and graph homomorphisms. We introduce the notion of a basis for morphisms of matroids, and show that its generating function is strongly log-concave. As a consequence, we obtain a generalization of Mason's conjecture on the f-vectors of independent subsets of matroids to arbitrary morphisms of matroids. To establish this, we define multivariate Tutte polynomials of morphisms of matroids, and show that they are Lorentzian in the sense of [6] for sufficiently small positive parameters.

P-Partitions and p-Positivity

Abstract Using the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse $P$-partitions expand positively into quasisymmetric power sums. Consequently, any nonnegative linear combination of such functions is $p$-positive whenever it is symmetric. As an application, we derive positivity results for chromatic quasisymmetric functions, unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials, and the more general $B$-polynomials, matroid quasisymmetric functions, and certain Eulerian quasisymmetric functions, thus reproving and improving on numerous results in the literature.

Linear Bound in Terms of Maxmaxflow for the Chromatic Roots of Series-Parallel Graphs

We prove that the (real or complex) chromatic roots of a series-parallel graph with maxmaxflow Lambda lie in the disc |q-1| < (Lambda-1)/log 2. More generally, the same bound holds for the (real or complex) roots of the multivariate Tutte polynomial when the edge weights lie in the "real antiferromagnetic regime" -1 \le v_e \le 0. This result is within a factor 1/log 2 \approx 1.442695 of being sharp

The multivariate arithmetic Tutte polynomial

We introduce an arithmetic version of the multivariate Tutte polynomial and a quasi-polynomial that interpolates between the two. A generalized Fortuin-Kasteleyn representation with applications to arithmetic colorings and flows is obtained. 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.

ARITHMETIC OF POTTS MODEL HYPERSURFACES

We consider Potts model hypersurfaces defined by the multivariate Tutte polynomial of graphs (Potts model partition function). We focus on the behavior of the number of points over finite fields for these hypersurfaces, in comparison with the graph hypersurfaces of perturbative quantum field theory defined by the Kirchhoff graph polynomial. We give a very simple example of the failure of the "fibration condition" in the dependence of the Grothendieck class on the number of spin states and of the polynomial countability condition for these Potts model hypersurfaces. We then show that a period computation, formally similar to the parametric Feynman integrals of quantum field theory, arises by considering certain thermodynamic averages. One can show that these evaluate to combinations of multiple zeta values for Potts models on polygon polymer chains, while silicate tetrahedral chains provide a candidate for a possible occurrence of non-mixed Tate periods.

Stability-to-instability transition in the structure of large-scale networks

We examine phase transitions between the "easy," "hard," and "unsolvable" phases when attempting to identify structure in large complex networks ("community detection") in the presence of disorder induced by network "noise" (spurious links that obscure structure), heat bath temperature T, and system size N. The partition of a graph into q optimally disjoint subgraphs or "communities" inherently requires Potts-type variables. In earlier work [Philos. Mag. 92, 406 (2012)], when examining power law and other networks (and general associated Potts models), we illustrated that transitions in the computational complexity of the community detection problem typically correspond to spin-glass-type transitions (and transitions to chaotic dynamics in mechanical analogs) at both high and low temperatures and/or noise. The computationally "hard" phase exhibits spin-glass type behavior including memory effects. The region over which the hard phase extends in the noise and temperature phase diagram decreases as N increases while holding the average number of nodes per community fixed. This suggests that in the thermodynamic limit a direct sharp transition may occur between the easy and unsolvable phases. When present, transitions at low temperature or low noise correspond to entropy driven (or "order by disorder") annealing effects, wherein stability may initially increase as temperature or noise is increased before becoming unsolvable at sufficiently high temperature or noise. Additional transitions between contending viable solutions (such as those at different natural scales) are also possible. Identifying community structure via a dynamical approach where "chaotic-type" transitions were found earlier. The correspondence between the spin-glass-type complexity transitions and transitions into chaos in dynamical analogs might extend to other hard computational problems. In this work, we examine large networks (with a power law distribution in cluster size) that have a large number of communities (q≫1). We infer that large systems at a constant ratio of q to the number of nodes N asymptotically tend towards insolvability in the limit of large N for any positive T. The asymptotic behavior of temperatures below which structure identification might be possible, T_{×}=O[1/lnq], decreases slowly, so for practical system sizes, there remains an accessible, and generally easy, global solvable phase at low temperature. We further employ multivariate Tutte polynomials to show that increasing q emulates increasing T for a general Potts model, leading to a similar stability region at low T. Given the relation between Tutte and Jones polynomials, our results further suggest a link between the above complexity transitions and transitions associated with random knots.

A motivic approach to phase transitions in Potts models

We describe an approach to the study of phase transitions in Potts models based on an estimate of the complexity of the locus of real zeros of the partition function, computed in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial. We give completely explicit calculations for the examples of the chains of linked polygons and of the graphs obtained by replacing the polygons with their dual graphs. These are based on a deletion–contraction formula for the Grothendieck classes and on generating functions for splitting and doubling edges.

Complex zero-free regions at large [formula omitted] for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights

We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) ZG(q,w) of a graph G with general complex edge weights w={we}. This generalizes a result of Sokal (2001) [28] that applies only within the complex antiferromagnetic regime |1+we|⩽1. Our proof uses the polymer-gas representation of the multivariate Tutte polynomial together with the Penrose identity.

Global disorder transition in the community structure of large-q Potts systems

We examine a global disorder transition when identifying community structure in an arbitrary complex network. Earlier, we illustrated (Hu D. et al., Philos. Mag., 92 (2012) 406) that “community detection” (CD) generally exhibits disordered (or unsolvable) and ordered (solvable) phases of both high and low computational complexity along with corresponding transitions from regular to chaotic dynamics in derived systems. Using an exact generalized dimensional reduction inequality, multivariate Tutte polynomials, and other considerations, we illustrate how increasing the number of communities q emulates increasing the heat bath temperature T for a general weighted Potts model, leading to global disorder in the community structure of arbitrary large graphs. Dimensional reduction bounds lead to results similar to those suggested by mean-field–type approaches. Large systems tend toward global insolvability in the limit of large q above a crossover temperature where |Je| is a typical interaction strength, L is the number of edges, and N is the number of nodes. For practical system sizes, a solvable phase is generally accessible at low T. The global nature of the disorder transition does not preclude solutions by local CD algorithms (even those that employ global cost function parameters) as long as community evaluations are locally determined.

Galois groups of multivariate Tutte polynomials

The multivariate Tutte polynomial $\hat Z_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure of $M$. Let $\bv = \{v_e\}_{e\in E}$, let $K$ be an arbitrary field, and suppose $M$ is connected. We show that $\hat Z_M$ is irreducible over $K(\bv)$, and give three self-contained proofs that the Galois group of $\hat Z_M$ over $K(\bv)$ is the symmetric group of degree $n$, where $n$ is the rank of $M$. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.

An Extension of Matroid Rank Submodularity and the $Z$-Rayleigh Property

We define an extension of matroid rank submodularity called $R$-submodularity, and introduce a minor-closed class of matroids called extended submodular matroids that are well-behaved with respect to $R$-submodularity. We apply $R$-submodularity to study a class of matroids with negatively correlated multivariate Tutte polynomials called the $Z$-Rayleigh matroids. First, we show that the class of extended submodular matroids are $Z$-Rayleigh. Second, we characterize a minor-minimal non-$Z$-Rayleigh matroid using its $R$-submodular properties. Lastly, we use $R$-submodularity to show that the Fano and non-Fano matroids (neither of which is extended submodular) are $Z$-Rayleigh, thus giving the first known examples of $Z$-Rayleigh matroids without the half-plane property.

A Zero-Free Interval for Chromatic Polynomials of Nearly 3-Connected Plane Graphs

Let be a nonseparable plane graph on vertices with at least two edges. Suppose that has outer face and that every 2-vertex-cut of contains at least one vertex of . Let denote the chromatic polynomial of . We show that for all . This result is a corollary of a more general result that for all , where is the multivariate Tutte polynomial of , , for all which are not incident to a vertex of , for all , for all other edges , and , are suitably chosen intervals with .

Expected Lengths of Minimum Spanning Trees for Non-identical Edge Distributions

An exact general formula for the expected length of the minimal spanning tree (MST) of a connected (possibly with loops and multiple edges) graph whose edges are assigned lengths according to independent (not necessarily identical) distributed random variables is developed in terms of the multivariate Tutte polynomial (alias Potts model). Our work was inspired by Steele's formula based on two-variable Tutte polynomial under the model of uniformly identically distributed edge lengths. Applications to wheel graphs and cylinder graphs are given under two types of edge distributions.

The multivariate signed Bollobás–Riordan polynomial

We generalise the signed Bollobás–Riordan polynomial of S. Chmutov and I. Pak [S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás–Riordan polynomial, Mos. Math. J. 7(3) (2007), 409–418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial, J. Combin. Theory, Ser. B 99(3) (2009), 617–638. arXiv:0711.3490, doi:10.1016/j.jctb.2008.09.007] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.

Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

The chromatic polynomial P G ( q ) of a loopless graph G is known to be non-zero (with explicitly known sign) on the intervals ( − ∞ , 0 ) , ( 0 , 1 ) and ( 1 , 32 / 27 ] . Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial Z G ( q , v ) . The proofs are quite simple, and employ deletion–contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27.

Tetromino tilings and the Tutte polynomial

We consider tiling rectangles of size 4m × 4n by T-shaped tetrominoes. Each tile is assigned a weight that depends on its orientation and position on the lattice. For a particular choice of the weights, the generating function of tilings is shown to be the evaluation of the multivariate Tutte polynomial ZG(Q, v) (known also to physicists as the partition function of the Q-state Potts model) on an (m − 1) × (n − 1) rectangle G, where the parameter Q and the edge weights v can take arbitrary values depending on the tile weights.

Around matrix-tree theorem

Generalizing the classical matrix-tree theorem we provide a for- mula counting subgraphs of a given graph with a fixed 2-core. We use this gen- eralization to obtain an analog of the matrix-tree theorem for the root system Dn (the classical theorem corresponds to the An-case). Several byproducts of the developed technique, such as a new formula for a specialization of the multivariate Tutte polynomial, are of independent interest.