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 nonzero (with explicitly known sign) on the intervals (-\infty,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.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure

The multivariate Tutte polynomial (alias Potts model) for graphs and matroids

The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial -- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial -- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics

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

A Little Statistical Mechanics for the Graph Theorist

In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches. This paper is an elementary general audience survey, intended to popularize the area and provide an accessible first point of entry for further exploration.Comment: 30 pages, 3 figure

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” $-1 \le v_e \le 0$. For each $\Lambda \geq 3$, we exhibit a family of graphs, namely, the “leaf-joined trees”, with maxmaxflow $\Lambda$ and chromatic roots accumulating densely on the circle $|q-1|=\Lambda -1$, thereby showing that our result is within a factor $1/\log 2 \approx 1.442695$ of being sharp