A simple decomposition for graphs yields generating functions for counting graphs by edges and connected components. A change of variables gives a new interpretation to the Tutte polynomial of the complete graph involving inversions of trees. The relation between the Tutte polynomial of the complete graph and the inversion enumerator for trees is generalized to the Tutte polynomial of an arbitr...

It is a classical result of Jaeger, Vertigan and Welsh that evaluating the Tutte polynomial of a graph is #P -hard in all but few special points. On the other hand, several papers in past years have shown that the Tutte polynomial of a graph can be efficiently computed for graphs of bounded tree-width. In this paper we present a recursive formula computing the Tutte polynomial of a matroid M re...

We find the universal module w(M) for functions (that need not be invariants) of finite matroids, defined on a minor-closed class M and with values in any module L over any commutative and unitary ring, that satisfy a parametrized deletion-contraction identity, F (M) = δeF (M r e) + γeF (M/e), when e is neither a loop nor a coloop. (F is called a (parametrized) weak Tutte function.) Within the ...

A strong Tutte function of matroids is a function of finite matroids which satisfies F ( M 1$M2) = F ( M 1 ) F ( M 2 ) and F ( M ) = aeF(M\e) + b e F ( M / e ) for e not a loop or coloop of M ,where ae , be are scalar parameters depending only on e . We classify strong Tutte functions of all matroids into seven types, generalizing Brylawski's classification of Tutte-Grothendieck invariants. One...

GraphsG andH are T -equivalent if they have the same Tutte polynomial. G is T -unique if any arbitrary graph H being T -equivalent to G implies that H is isomorphic to G. We show that the generalized Petersen graph P (m, 2) and the line graph of P (m, 2) are T -unique. 1 The Tutte Polynomial Let G be a graph with vertex set V and edge set E, and let G[A] denote the subgraph ofG induced by the e...

We observe that a formula given by S. Negami [Trans. Amer. Math. Soc. 299 (1987), 601-622] for the Tutte polynomial of a k -sum of two graphs generalizes to a colored Tutte polynomial. Consequently an algorithm of A. Andrzejak [Discrete Math. 190 (1998), 39-54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has als...

The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded tree-width. The notion of clique-width extends the definition of cograhs (graphs without induced P4), and it is a more general notion than that of tree-width. We show a subexponential algorithm (running in time exp O(n) ) for compu...

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V (G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here ...

We consider a specialization YM (q, t) of the Tutte polynomial of a matroid M which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of M . We show that the coefficients of YM (1 − p, t) are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independe...