A weighted graph polynomial from chromatic invariants of knots

Noble, Steven D.; Welsh, Dominic

doi:10.5802/aif.1706

Cited by 86 publications

(126 citation statements)

References 16 publications

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“…In [Mak04] the second author has introduced the MSOL-definable and the SOL-definable graph polynomials, the class of graph polynomials where the range of summation is definable in (monadic) second order logic. He has verified that all the examples of graph polynomials discussed in the literature, with the exception of the weighted graph polynomial of [NW99], are actually SOL-polynomials over some expansions (by adding order relations) of the graph, cf. also [Mak06].…”

Section: Graph Invariants and Graph Polynomialsmentioning

confidence: 96%

On Counting Generalized Colorings

Kotek

Makowsky

Zilber

2008

Computer Science Logic

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Abstract. The notion of graph polynomials definable in Monadic Second Order Logic, MSOL, was introduced in [Mak04]. It was shown that the Tutte polynomial and its generalization, as well as the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. In this paper we present a framework of graph polynomials based on counting functions of generalized colorings. We show that this class encompasses the examples of graph polynomials from the literature. Furthermore, we extend the definition of graph polynomials definable in MSOL to allow definability in full second order, SOL. Finally, we show that the SOL-definable graph polynomials extended with a combinatorial counting function are exactly the counting functions of generalized colorings definable in SOL.

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Section: Graph Invariants and Graph Polynomialsmentioning

confidence: 96%

On Counting Generalized Colorings

Kotek

Makowsky

Zilber

2008

Computer Science Logic

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“…The weighted graph polynomial from [NW99], however, is not invariant under variable renaming because the integer index of the variables carries a graph theoretic meaning. It is this feature which allows us to show that U (G,X, Y ) is not SOL-definable.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…In Section 7 we discuss a graph polynomial introduced in [NW99], which is provably not a SOL-definable graph polynomial. It is defined by a subset expansion formula, where the exponents fi(G,Ā) depend on i, which is not allowed in our definition of SOL-definable graph polynomials.…”

Section: Introductionmentioning

confidence: 99%

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Graph Polynomials: From Recursive Definitions to Subset Expansion Formulas

Godlin¹,

Katz²,

Makowsky³

2010

Journal of Logic and Computation

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Abstract. Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this paper we present a general, logic-based framework which gives a precise meaning to recursive definitions of graph polynomials. We then prove that in this framework every recursive definition of a graph polynomial can be converted into a subset expansion formula.

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“…Graphs with integers attached to each vertex and their corresponding multivariate polynomials W G .˛e; N v / have been considered in [50]. But to represent external variables we need to replace the integer N v by a set of N v disjoint objects, 1 hereafter called flags (see Section 2.1).…”

Section: Decorated Graphsmentioning

confidence: 99%

Topological graph polynomials and quantum field theory Part I: heat kernel theories

Krajewski¹,

Rivasseau²,

Tanasă³

et al. 2010

J. Noncommut. Geom.

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Abstract. We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first article we consider translation invariant theories with the usual heat-kernel-based propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are special versions of the Bollobás-Riordan polynomials.Mathematics Subject Classification (2010). 81T18, 05C31, 05C05, 05C10.

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A weighted graph polynomial from chromatic invariants of knots

Cited by 86 publications

References 16 publications

On Counting Generalized Colorings

On Counting Generalized Colorings

Graph Polynomials: From Recursive Definitions to Subset Expansion Formulas

Topological graph polynomials and quantum field theory Part I: heat kernel theories

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