About: The Potts model and chromatic functions of graphs

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
rdfs:seeAlso	http://www.ems-ph.org/journals/journal.php?jrn=aihpd
Description	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. 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. (en)
Title	The Potts model and chromatic functions of graphs The Potts model and chromatic functions of graphs (en)
skos:prefLabel	The Potts model and chromatic functions of graphs The Potts model and chromatic functions of graphs (en)
skos:notation	RIV/00216208:11320/14:10286492!RIV15-GA0-11320___
http://linked.open...avai/riv/aktivita	P
http://linked.open...avai/riv/aktivity	P(GBP202/12/G061)
http://linked.open...iv/cisloPeriodika	1
http://linked.open...vai/riv/dodaniDat	2015
http://linked.open...aciTvurceVysledku	Klazar, Martin Loebl, Martin
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	38170
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/14:10286492
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	graph; chromatic function; Potts model (en)
http://linked.open.../riv/klicoveSlovo	graph Potts model chromatic function
http://linked.open...odStatuVydavatele	CH - Švýcarská konfederace
http://linked.open...ontrolniKodProRIV	[35D578774BDA]
http://linked.open...i/riv/nazevZdroje	Annales de l'Institut Henri Poincaré D. Combinatorics, Physics and their Interactions
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	2 (xsd:int)
http://linked.open...cetTvurcuVysledku	3 (xsd:int)
http://linked.open...vavai/riv/projekt	Center of excellence - Institute for theoretical computer science (CE-ITI)
http://linked.open...UplatneniVysledku	2014
http://linked.open...v/svazekPeriodika	1
http://linked.open...iv/tvurceVysledku	Loebl, Martin Klazar, Martin Moffatt, Iain
issn	2308-5827
number of pages	14 (xsd:int)
http://bibframe.org/vocab/doi	10.4171/AIHPD/2
http://localhost/t...ganizacniJednotka	11320

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