Graph polynomials and Tutte-Grothendieck invariants: an application of elementary finite Fourier analysis

Strong Tutte Functions of Matroids and Graphs

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...

Tutte Polynomials of Flower Graphs

The arithmetic Tutte polynomials of the classical root systems

Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems An, Bn, Cn, and Dn with respect to their integer, root, and weight lattices. We do ...

Tutte polynomials of hyperplane arrangements and the finite field method

The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement, which answers a wide variety of questions about its underlying object. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a h...

A Note on Jacobians, Tutte Polynomials, and Two-Variable Zeta Functions of Graphs

Address correspondence to Sam Payne, Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06511, USA. Email: [email protected] We address questions posed by Lorenzini about relations between Jacobians, Tutte polynomials, and the Brill–Noether theory of finite graphs, as encoded in his two-variable zeta functions. In particular, we give examples showing that none of th...