3,150 research outputs found
Homomorphisms and polynomial invariants of graphs
This paper initiates a general study of the connection between graph homomorphisms and the Tutte
polynomial. This connection can be extended to other polynomial invariants of graphs related to the Tutte
polynomial such as the transition, the circuit partition, the boundary, and the coboundary polynomials.
As an application, we describe in terms of homomorphism counting some fundamental evaluations of the
Tutte polynomial in abelian groups and statistical physics. We conclude the paper by providing a
homomorphism view of the uniqueness conjectures formulated by Bollobás, Pebody and Riordan.Ministerio de Educación y Ciencia MTM2005-08441-C02-01Junta de Andalucía PAI-FQM-0164Junta de Andalucía P06-FQM-0164
Graph homomorphisms, the Tutte polynomial and “q-state Potts uniqueness”
We establish for which weighted graphs H homomorphism functions from multigraphs
G to H are specializations of the Tutte polynomial of G, answering a question
of Freedman, Lov´asz and Schrijver.
We introduce a new property of graphs called “q-state Potts uniqueness” and relate
it to chromatic and Tutte uniqueness, and also to “chromatic–flow uniqueness”,
recently studied by Duan, Wu and Yu.Ministerio de Educación y Ciencia MTM2005-08441-C02-0
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 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 and and compare this with the
corresponding minimal examples for the -polynomial, which generalizes the
well known Tutte-polynomal.Comment: V2. Extended versio
Distinguishing graphs by their left and right homomorphism profiles
We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte
uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu.
We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are
specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and
Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from
multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and
Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky.
Unifying the study of these and related problems is the notion of the left and right homomorphism profiles
of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164
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