We introduce a polynomial invariant of graphs on surfaces, PG , generalizing the classical Tutte polynomial. Poincaré duality on surfaces gives rise to a natural duality result for PG , analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where Tutte polynomial is known as the partition function of the Pott...

Bollobás and Riordan introduce a Tutte polynomial for coloured graphs and matroids in [3]. We observe that this polynomial has an expansion as a sum indexed by the subsets of the ground-set of a coloured matroid, generalizing the subset expansion of the Tutte polynomial. We also discuss similar expansions of other contraction–deletion invariants of graphs and matroids. 1. The coloured Tutte pol...

We present two splitting formulas for calculating the Tutte polynomial of a matroid. The rst one is for a generalized parallel connection across a 3-point line of two matroids and the second one is applicable to a 3-sum of two matroids. An important tool used is the bipointed Tutte polynomial of a matroid, an extension of the pointed Tutte polynomial introduced by Thomas Brylawski in Bry71].

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [ABS00] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [ABS04b], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-null...

The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name B-polynomial. The B-polynomial has three variables, but when specialized to the case of graphs (that is, digraphs where arcs come in pairs with opposite directions), one of the variables becomes redundant and the B-polynomial ...

The coloured Tutte polynomial by Bollobás and Riordan is, as a generalisation of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contractiondeletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #P-hard to evaluate almost everywhere by establishing reductions along curves and lines. We establish a similar...

This paper is based on a series of talks given at the Patejdlovka Enumeration Workshop held in the Czech Republic in November 2007. The topics covered are as follows. The graph polynomial, Tutte-Grothendieck invariants, an overview of relevant elementary finite Fourier analysis, the Tutte polynomial of a graph as a Hamming weight enumerator of its set of tensions (or flows), description of a fa...