The Martin polynomial of an oriented Eulerian graph encodes information about families of cycles in the graph. This paper uses a transformation of the Martin polynomial that facilitates standard combinatorial manipulations. These manipulations result in several new identities for the Martin polynomial, including a di.erentiation formula. These identities are then applied to get new combinatoria...

The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion-contraction relations, its specialisation to cellularly embedded graphs does not. Here we ex...

We give a quantum-inspired Opnq algorithm computing the Tutte polynomial of a lattice path matroid, where n is the size of the ground set of the matroid. Furthermore, this can be improved to Opnq arithmetic operations if we evaluate the Tutte polynomial on a given input, fixing the values of the variables. The best existing algorithm, found in 2004, was Opnq, and the problem has only been known...

For any graph G with n edges, the spanning subgraphs and the orientations of G are both counted by the evaluation TG(2, 2) = 2 n of its Tutte polynomial. We define a bijection Φ between spanning subgraphs and orientations and explore its enumerative consequences regarding the Tutte polynomial. The bijection Φ is closely related to a recent characterization of the Tutte polynomial relying on a c...

The Tutte polynomial of a graph, also known as the partition function of the q-state Potts model, is a 2-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chromatic polynomial and flow polynomial as partial evaluations, and various numerical invariants, such as the number of ...

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, there are large regions of the parameter space for which computing the sign of the Tutte polynomial is actually #P-hard. As a trivial cons...

We present several new polynomial identities associated with matroids and geometric lattices, and relate them to formulas for the characteristic polynomial and the Tutte polynomial. The identities imply a formula for the L^ e numbers of complex hyperplane arrangements.

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, there are large regions of the parameter space for which computing the sign of the Tutte polynomial is actually #P-hard. As a trivial cons...

The recently introduced chain and sheaf polynomials of a graph are shown to be essentially equivalent to a weighted version of the Tutte polynomial. c © 2002 Elsevier Science B.V. All rights reserved.

The Tutte polynomial TG(X,Y ) of a graph G is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs G , TG(X,Y ) = TG∗(Y,X) where G∗ denotes the dual graph. We examine this property from the perspective of manifold topology, formulating polynomial invariants for higher-dimensional simplicial com...