The Tutte Polynomial and Its Applications

Algebraic and Geometric Methods in Enumerative Combinatorics

“…The wonderful surveys [BO92], [EMM11], and [Wel93] include most of the results mentioned here; we provide references for the other ones.…”

Computing the Tutte polynomial of Archimedean tilings

“…A function F from (isomorphism classes of) graphs to the polynomial ring C½a; b; c; x; y is a generalized Tutte-Gröthen-dieck invariant [4,11] if it satisfies, for each graph G ¼ ðV; EÞ and any edge e 2 E, the following recurrence relations…”

An inequality for Tutte polynomials

“…A proof of this lemma together with further information on Tutte polynomials can also be found in the text book by Tutte [8], and in the survey article of Brylawski and Oxley [1]. We may use Lemma 1 and induction to deduce that the Tutte polynomial of a graph with at least one edge has nonnegative coefficients and zero constant term.…”

“…The Tutte polynomial of a graph contains a large amount of information about the graph. It follows from equation (1) that T G (1, 1) is equal to the number of spanning trees of a connected graph G. Stanley [6] showed that T G (2, 0) is the number of acyclic orientations of G. It also follows from either Greene and Zaslavsky [3] or Las Vergnas [4], see [1,Example 6.3.8], that T G (0, 2) is the number of totally cyclic orientations of G i.e. orientations in which every arc lies in a directed cycle.…”