Multivariable, parameterized, and colored extensions of the Tutte polynomial

Abstract

Multivariable and paramaterized versions of various forms of the Tutte polynomial arise from assigning parameters to the edges of a graph or to the elements of the ground set of a matroid. These parameters can capture combinatorial properties such as polynomial computation trees and series-parallel reductions, as well as information relevant to applications such as statistical mechanics models and resistances of circuit elements. This chapter covers several forms of these generalizations. A general form for doubly parameterized Tutte polynomials. Formulas for duality, direct sums, and various types of edges (loops, bridges, series-parallel, etc.). Equivalence with the cycle matroid. Deletion–contraction form and an activities expansion. The colored Tutte polynomial and strong Tutte functions. Ported polynomials, the relative Tutte polynomial, knot invariants, and variations.