Evaluating a weighted graph polynomial for graphs of bounded tree-width

We show that for any $k$ there is a polynomial time algorithm to evaluate the weighted graph polynomial $U$ of any graph with tree-width at most $k$ at any point. For a graph with $n$ vertices, the algorithm requires $O(a_k n^{2k+3})$ arithmetical operations, where $a_k$ depends only on $k$

Evaluating the rank generating function of a graphic 2-polymatroid

We consider the complexity of the two-variable rank generating function, $S$, of a graphic 2-polymatroid. For a graph $G$, $S$ is the generating function for the number of subsets of edges of $G$ having a particular size and incident with a particular number of vertices of $G$. We show that for any $x,y \in \mathbb{Q}$ with $xy \not = 1$, it is $\#$P-hard to evaluate $S$ at $(x,y)$. We also consider the $k$-thickening of a graph and computing $S$ for the $k$-thickening of a graph

Potts models with magnetic field: arithmetic, geometry, and computation

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that become polynomially countable after a perturbation of the magnetic field. The same recursive formula for the Grothendieck classes, under edge-doubling operations, holds as in the case without magnetic field, but the closed formulae for specific examples like banana graphs differ in the presence of magnetic field. We give examples of computation of the Euler characteristic with compact support, for the set of real zeros, and find a similar exponential growth with the size of the graph. This can be viewed as a measure of topological and algorithmic complexity. We also consider the computational complexity question for evaluations of the polynomial, and show both tractable and NP-hard examples, using dynamic programming.Comment: 16 pages, LaTeX; v2: final version with small correction

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function $XB$, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of $XB$ and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted $XB$ admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting $XB$ to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.Comment: 28 page

Complexity of Ising Polynomials

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by L. A. Goldberg, M. Jerrum and M. Paterson in 2003. Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied by D. Andr\'{e}n and K. Markstr\"{o}m in 2009. We consider the complexity of Z(G;t,y) and Z(G;x,y,z) in comparison to that of the Tutte polynomial, which is well-known to be closely related to the Potts model in the absence of an external field. We show that Z(G;\x,\y,\z) is #P-hard to evaluate at all points in $mathbb{Q}^3$, except those in an exception set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by H. Dell, T. Husfeldt and M. Wahl\'{e}n in 2010 in order to study the complexity of the Tutte polynomial. In analogy to their results, we give a dichotomy theorem stating that evaluations of Z(G;t,y) either take exponential time in the number of vertices of $G$ to compute, or can be done in polynomial time. Finally, we give an algorithm for computing Z(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial

Evaluating a Weighted Graph Polynomial for Graphs of Bounded Tree-Width

We show that for any k there is a polynomial time algorithm to evaluate the weighted graph polynomial U of any graph with tree-width at most k at any point. For a graph with n vertices, the algorithm requires O ( a k n 2 k +3 ) arithmetical opera- tions, where a k depends only on