The Polychromate and a Chord Diagram Polynomial

Sarmiento, Irasema

doi:10.1007/s000260050008

Cited by 24 publications

(22 citation statements)

References 12 publications

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“…Note however that there are non-isomorphic graphs with the same U polynomial. This is a corollary of a result of Sarmiento [26], showing that the coefficients of U and the polychromate determine one another. It remains an open problem to determine whether or not there are two nonisomorphic trees with the same U polynomial.…”

Section: Introductionsupporting

confidence: 52%

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

Noble

2009

Electron. J. Combin.

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Section: Introductionsupporting

confidence: 52%

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

Noble

2009

Electron. J. Combin.

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“…Sarmiento proved in [7] that the U -polynomial, and hence XB, is also equivalent to Brylawski's polychromate from [1]. In [2], it was shown that U (G) has a close connection with the Potts model (see Section 3.2 for a discussion of the Potts model): for x i = ( k j=1 e iβHj )/(e βJ − 1), We now turn our attention to our second family of graph polynomials.…”

Section: Introductionmentioning

confidence: 95%

The Potts model and chromatic functions of graphs

Klazar¹,

Loebl²,

Moffatt³

2014

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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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 Bq(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 Br,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 Br,q(G; x, k) is equivalent to the U -polynomial and to Stanley's symmetric bad colouring polynomial.This completes the proof of the claim, and hence B r,q determines XB, completing the proof of the theorem.As a corollary we obtain the equivalence of B r,q and U .Corollary 2. Let r > 1 be given. Then graph functions B r,q and U are equivalent.Proof. This follows from Theorem 1, and Theorem 6.2 of [6] which gives the equivalence of U and XB.

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“…In [NW99] it was also proven that the U -polynomial is equivalent to the Tutte symmetric function introduced by Stanley in [Sta98]. Tightly following this Sarmiento [Sar00] proved that the U -polynomial, and hence the Tutte symmetric function as well, is equivalent to the so called polychromate, introduced by Brylawski [Bry81] in 1981. Brylawski also proved that the polychromate of G determines the polychromate of G, and hence the same is true for the U -polynomial and the Tutte symmetric function.…”

Section: Families Of Graphs With the Same Homomorphism Polynomialmentioning

confidence: 91%

From the Ising and Potts Models to the General Graph Homomorphism Polynomial

Markström¹

2016

Graph Polynomials

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The Polychromate and a Chord Diagram Polynomial

Cited by 24 publications

References 12 publications

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

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

The Potts model and chromatic functions of graphs

From the Ising and Potts Models to the General Graph Homomorphism Polynomial

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