Some new evaluations of the Tutte polynomial

Goodall, Andrew

doi:10.1016/j.jctb.2005.07.007

Cited by 10 publications

(12 citation statements)

References 7 publications

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“…We can now state and prove the Cheshire-cat identity. This identity generalizes identities in [6,13].…”

Section: The Cheshire-cat Identitysupporting

confidence: 58%

Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

Kung

2012

Journal of Combinatorial Theory, Series B

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Let G be a matrix and M(G) be the matroid defined by linear dependence on the set E of column vectors of G. Roughly speaking, a parcel is a subset of pairs ( f , g) of functions defined on E to a suitable Abelian group A satisfying a coboundary condition (that the difference f − g is a flow over A of G) and a congruence condition (that an algebraic or combinatorial function of f and g, such as the sum of the size of the supports of f and g, satisfies some congruence condition). We prove several theorems of the form: a linear combination of sizes of parcels, with coefficients roots of unity, equals a multiple of an evaluation of the Tutte polynomial of M(G) at a point (u, v), usually with complex coordinates, satisfyingAn aim of the graph-theoretic results is to reformulate theorems, like the 4-color theorem, in probabilistic terms. The statements of these results involve differences in probabilities and their proofs are intricate, involving commutative algebra or finite Fourier analysis. Our aim in this paper is different: it is to put these results in context, as initial cases of infinite families of results about sizes of parcels constructed using flows of matrices over suitable Abelian groups.This paper is the second in a series. The earlier paper [13] is about the "parametric" theory. Using characters, we studied parcels defined by algebraic conditions using actual values of elements in the Abelian group A. With two exceptions, the present paper is about the "non-parametric" theory. Parcels are defined using congruence conditions which only use the data whether an element in A is zero or nonzero. We will use non-parametric parcels to obtain combinatorial interpretations of evaluations of rank generating polynomials on complex hyperbolas λx = q, where q is an integer greater than 1.(There seems to be only one interpretation of the evaluation of the rank generating polynomial at complex points in the literature, the interpretation by Jaeger [11] of the value of the Tutte polynomial at (e 2π ι/3 , e −2πι/3 ). Jaeger's interpretation will be discussed in Section 6.)We shall assume some knowledge of graph and matroid theory. See, for example, [1,12,16]. We recall briefly concepts and definitions central to this paper. Let G be a matrix with columns indexed by the set E and M(G) be the matroid on E defined by linear dependence of the column vectors.Let Γ (V , E) be a graph with a chosen orientation on the edges. The orientation gives a function sign(v, e): if e is not a loop, then sign(v, e) equals +1, −1, or 0, depending on whether the edge e is going into v, going out of v, or not incident at all on v. If e is a loop, then sign(v, e) is always 0. We will use two matrices defined by Γ . The first is the vertex-edge matrix, the |V | × |E| matrix H with rows indexed by V and columns indexed by E such that the ve-entry is sign(v, e). The matroid on E defined by the vertex-edge matrix is the cycle matroid of Γ . The second is the cycle-edge matrix. We think of a cycle c as a set of edges e 0 , e 1 , . . . , e t−1 such...

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“…We can now state and prove the Cheshire-cat identity. This identity generalizes identities in [6,13].…”

Section: The Cheshire-cat Identitysupporting

confidence: 58%

Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

Kung

2012

Journal of Combinatorial Theory, Series B

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“…From this we can derive Matiyasevich's necessary condition in terms of equivalence classes of such orientations. Analogously, it should be possible to derive the similar result [Go,Theorem 18] by Goodall. Moreover, if we apply part (v), with b ≡= 1 , to the line graph of a plane triangulation we further can deduce [Ma2,Theorem 7].…”

Section: Colorings Of Graphsmentioning

confidence: 83%

Colorings and Nowhere-Zero Flows of Graphs in Terms of Berlekamp's Switching Game

Schauz

2011

Electron. J. Combin.

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We work with a unifying linear algebra formulation for nowhere-zero flows and colorings of graphs and matrices. Given a subspace (code) $U\leq{\mathbb{Z}_k^n}$ – e.g. the bond or the cycle space over ${\mathbb{Z}}_k$ of an oriented graph – we call a nowhere-zero tuple $f\in{\mathbb{Z}_k^n}$ a flow of $U$ if $f$ is orthogonal to $U$. In order to detect flows, we view the subspace $U$ as a light pattern on the $n$-dimensional Berlekamp Board ${\mathbb{Z}_k^n}$ with $k^n$ light bulbs. The lights corresponding to elements of $U$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. The core result of this paper is that the subspace $U$ has a flow if and only if the light pattern $U$ cannot be switched off. In particular, a graph $G$ has a nowhere-zero $k$-flow if and only if the ${\mathbb{Z}}_k$-bond space of $G$ cannot be switched off. It has a vertex coloring with $k$ colors if and only if a certain corresponding code over ${\mathbb{Z}}_k$ cannot be switched off. Similar statements hold for Tait colorings, and for nowhere-zero points of matrices. Studying different normal forms to equivalence classes of light patterns, we find various new equivalents, e.g., for the Four Color Problem, Tutte's Flow Conjectures and Jaeger's Conjecture. Two of our equivalents for colorability and existence of nowhere zero flows of graphs include as special cases results by Matiyasevich, by Balázs Szegedy, and by Onn. Alon and Tarsi's sufficient condition for $k$-colorability also arrives, remarkably, as a generalized full equivalent.

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“…[Wel93]. All these are also graph parameters which take values in N. More sophisticated evaluations of the Tutte polynomial can be found in [Goo06,Goo08].…”

Section: Graph Parameters and Graph Polynomialsmentioning

confidence: 99%

Evaluations of Graph Polynomials

Godlin

Kotek

Makowsky

2008

Graph-Theoretic Concepts in Computer Science

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Abstract.A graph polynomial p(G,X) can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific pointsX =x0. In this paper we study the question how to prove that a given graph parameter, say ω(G), the size of the maximal clique of G, cannot be a fixed coefficient or the evaluation at any point of the Tutte polynomial, the interlace polynomial, or any graph polynomial of some infinite family of graph polynomials.Our result is very general. We give a sufficient condition in terms of the connection matrix of graph parameter f (G) which implies that it cannot be the evaluation of any graph polynomial which is invariantly definable in CMSOL, the Monadic Second Order Logic augmented with modular counting quantifiers. This criterion covers most of the graph polynomials known from the literature.

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Some new evaluations of the Tutte polynomial

Cited by 10 publications

References 7 publications

Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

Colorings and Nowhere-Zero Flows of Graphs in Terms of Berlekamp's Switching Game

Evaluations of Graph Polynomials

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