A graph polynomial is an algebraic object associated with a graph that is usually invariant at least under graph isomorphism. As such, it encodes information about the graph, and enables algebraic methods for extracting this information. This chapter surveys a comprehensive, although not exhaustive, sampling of graph polynomials. It concludes Graph Polynomials and their Applications I: The Tutt...

for α, γ ∈ {0, 1, 2} and β , δ ∈ {0, 1}. We introduce an intersection lattice of 64 cut–cycle fourientation classes enumerated by generalized Tutte polynomial evaluations of this form. We prove these enumerations using a single deletion–contraction argument and classify axiomatically the set of fourientation classes to which our deletion–contraction argument applies. This work unifies and exten...

We define and study the Tutte polynomial of a hyperplane arrangement. We introduce a method for computing it by solving an enumerative problem in a finite field. For specific arrangements, the computation of Tutte polynomials is then reduced to certain related enumerative questions. As a consequence, we obtain new formulas for the generating functions enumerating alternating trees, labelled tre...

We introduce a multiplicity Tutte polynomial M(x, y), with applications to zonotopes and toric arrangements. We prove that M(x, y) satisfies a deletion-restriction recursion and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x, y), likewise the corresponding polynomials fo...

We give a new characterization of the Tutte polynomial of graphs. Our characterization is formally close (but inequivalent) to the original definition given by Tutte as the generating function of spanning trees counted according to activities. Tutte’s notion of activity requires a choice of a linear order on the edge set (though the generating function of the activities is, in fact, independent...

We define two two-variable polynomials for rooted trees and one twovariable polynomial for unrooted trees, all of which are based on the coranknullity formulation of the Tutte polynomial of a graph or matroid. For the rooted polynomials, we show that the polynomial completely determines the rooted tree, i.e., rooted trees TI and T, are isomorphic if and only if f(T,) = f(T2). The corresponding ...