Comparative Theory for Graph Polynomials (Dagstuhl Seminar 19401) (original) (raw)
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Evaluations of Graph Polynomials
2008
A graph polynomial p(G,barX)p(G, \bar{X})p(G,barX) 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 points barX=barx_0\bar{X}= \bar{x}_0barX=barx_0 . 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.
2015
In this paper we observe the problem of counting graph colorings using polynomials. Several reformulations of The Four Color Conjecture are considered (among them algebraic, probabilistic and arithmetic). In the last section Tutte polynomials are mentioned. 1.
The Tutte polynomial of some matroids
2014
The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any mul-tiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engi-neering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of graphs or matroids. In this work we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were use to find the for-mulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.
Graph Polynomials: From Recursive Definitions to Subset Expansion Formulas
Journal of Logic and Computation, 2010
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.
Graph polynomials and symmetries
Journal of Algebra and Its Applications, 2018
In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.
A most general edge elimination graph polynomial
2007
We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and deletion of edges together with their end points. Like in the case of deletion and contraction only (W. Tutte, 1954), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call xi(G,x,y,z)\xi(G,x,y,z)xi(G,x,y,z). We show that the new polynomial simultaneously generalizes the Tutte polynomial, the matching polynomial, and the recent generalization of the chromatic polynomial proposed by K.Dohmen, A.P\"{o}nitz and P.Tittman (2003), including also the independent set polynomial of I. Gutman and F. Harary, (1983) and the vertex-cover polynomial of F,M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little (2002). We establish two definitions of the new polynomial: first, the most general confluent recursive definition, and then an explicit one, using a set expansion formula, and prove their identity. We further expand this result to edge-labeled graphs as was done for the Tutte polynomial by T. Zaslavsky (1992) and B. Bollob\'as and O. Riordan (1999). The edge labeled polynomial xilab(G,x,y,z,bart)\xi_{lab}(G,x,y,z, \bar{t})xilab(G,x,y,z,bart) also generalizes the chain polynomial of R.C. Read and E.G. Whitehead Jr. (1999). Finally, we discuss the complexity of computing xi(G,x,y,z)\xi(G,x,y,z)xi(G,x,y,z).
The Equivalence of Two Graph Polynomials and a Symmetric Function
Combinatorics, Probability and Computing, 2009
The U -polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial due to Stanley are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to generalize them which also captures Tutte's universal V -functions as a specialization. We show that the equivalence remains true for the extended functions thus answering a question raised by Dominic Welsh.
Polynomial invariants of graphs with state models
Discrete Applied Mathematics, 1995
We reformulate a polynomial invariant of graphs defined by Negami, using the notion of state models, and discuss another polynomial invariant, as a natural extension of Negami's polynomial, which can distinguish many graphs more finely than the original.