The Tutte Polynomial and Its Applications

“…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…”

Section: The Tutte Polynomialmentioning

confidence: 99%

Computing the Tutte polynomial of Archimedean tilings

Garijo

Gegúndez

Márquez

et al. 2014

Applied Mathematics and Computation

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We describe an algorithm to compute the Tutte polynomial of large fragments of Archimedean tilings by squares, triangles, hexagons and combinations thereof. Our algorithm improves a well known method for computing the Tutte polynomial of square lattices. We also address the problem of obtaining Tutte polynomial evaluations from the symbolic expressions generated by our algorithm, improving the best known lower bound for the asymptotics of the number of spanning forests, and the lower and upper bounds for the asymptotics of the number of acyclic orientations of the square lattice.

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“…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.…”

Section: Introductionmentioning

confidence: 98%

“…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.…”

Section: Introductionmentioning

confidence: 99%