The equivalence of two graph polynomials and a symmetric function

The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial due to Stanley are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to strengthen them which also captures Tutte's universal V-function as a specialization. We show that the equivalence remains true for the strong functions thus answering a question raised by Dominic Welsh

The Equivalence of Two Graph Polynomials and a Symmetric Function

The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial due to Stanley are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to generalize them which also captures Tutte's universal V-functions as a specialization. We show that the equivalence remains true for the extended functions thus answering a question raised by Dominic Welsh.Comment: 17 page

A combinatorial proof that Schubert vs. Schur coefficients are nonnegative

We give a combinatorial proof that the product of a Schubert polynomial by a Schur polynomial is a nonnegative sum of Schubert polynomials. Our proof uses Assaf's theory of dual equivalence to show that a quasisymmetric function of Bergeron and Sottile is Schur-positive. By a geometric comparison theorem of Buch and Mihalcea, this implies the nonnegativity of Gromov-Witten invariants of the Grassmannian.Comment: 26 pages, several colored figure

From the Ising and Potts models to the general graph homomorphism polynomial

In this note we study some of the properties of the generating polynomial for homomorphisms from a graph to at complete weighted graph on $q$ vertices. We discuss how this polynomial relates to a long list of other well known graph polynomials and the partition functions for different spin models, many of which are specialisations of the homomorphism polynomial. We also identify the smallest graphs which are not determined by their homomorphism polynomials for $q=2$ and $q=3$ and compare this with the corresponding minimal examples for the $U$-polynomial, which generalizes the well known Tutte-polynomal.Comment: V2. Extended versio

The Potts model and chromatic functions of graphs

The $U$-polynomial of Noble and Welsh is known to have intimate connections with the Potts model as well as with several important graph polynomials. For each graph $G$, $U(G)$ is equivalent to Stanley's symmetric bad colouring polynomial $XB(G)$. Moreover Sarmiento established the equivalence between $U$ and the polychromate of Brylawski. Loebl defined the $q$-dichromate $B_q(G,x,y)$ as a function of a graph $G$ and three independent variables $q,x,y$, proved that it is equal to the partition function of the Potts model with variable number of states and with a certain external field contribution, and conjectured that the $q$-dichromate is equivalent to the $U$-polynomial. He also proposed a stronger conjecture on integer partitions. The aim of this paper is two-fold. We present a construction disproving Loebl's integer partitions conjecture, and we introduce a new function $B_{r,q}(G;x,k)$ which is also equal to the partition function of the Potts model with variable number of states and with a (different) external field contribution, and we show that $B_{r,q}(G;x,k)$ is equivalent to the $U$-polynomial and to Stanley's symmetric bad colouring polynomial