We calculate Jones polynomials VL(t) for several families of alternating knots and links by computing the Tutte polynomials T (G, x, y) for the associated graphs G and then obtaining VL(t) as a special case of the Tutte polynomial. For each of these families we determine the zeros of the Jones polynomial, including the accumulation set in the limit of infinitely many crossings. A discussion is ...

We show that evaluating the Tutte polynomial for the class of bicircular matroids is #Phard at every point (x, y) except those in the hyperbola (x − 1)(y − 1) = 1 and possibly those on the lines x = 0 and x = −1. Since bicircular matroids form a rather restricted subclass of transversal matroids, our results can be seen as a partial strengthening of a result by Colbourn, Provan and Vertigan, na...

Let G be a multigraph. We study the space X (G) of all pictures of G in complex projective d-space. The main result is that the homology groups (with integer coefficients) of X (G) are completely determined by the Tutte polynomial of G. One application is a criterion in terms of the Tutte polynomial for independence in the d-parallel matroids studied in combinatorial rigidity theory. In the cas...

The Tutte polynomial is an important concept in graph theory which captures many important properties of graphs (e.g. chromatic number, number of spanning trees etc). It also provides a normalised representation that can be used as an equivalence relation on graphs and has applications in diverse areas such micro-biology and physics. A highly efficient algorithm for computing Tutte polynomials ...

We develop constructive techniques to show that non-isomorphic 3-connected matroids that are representable over a fixed finite field and that have the same Tutte polynomial abound. In particular, for most prime powers q, we construct infinite families of sets of 3-connected matroids for which the matroids in a given set are non-isomorphic, are representable over GF(q), and have the same Tutte p...

An exact general formula for the expected length of the minimal spanning tree (MST) of a connected (possibly with loops and multiple edges) graph whose edges are assigned lengths according to independent (not necessarily identical) distributed random variables is developed in terms of the multivariate Tutte polynomial (alias Potts model). Our work was inspired by Steele’s formula based on two-v...

Using the decomposition theory of modular and integral flow polynomials, we answer a problem of Beck and Zaslavsky, by providing a general situation in which the integral flow polynomial is a multiple of the modular flow polynomial.

If Tn(x, y) is the Tutte polynomial of the complete graph Kn, we have the equality Tn+1(1, 0) = Tn(2, 0). This has an almost trivial proof with the right combinatorial interpretation of Tn(1, 0) and Tn(2, 0). We present an algebraic proof of a result with the same flavour as the latter: Tn+2(1,−1) = Tn(2,−1), where Tn(1,−1) has the combinatorial interpretation of being the number of 0–1–2 incre...

Let G be a connected graph with vertex set {0, 1, 2, . . . , n}. We allow G to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of G-parking functions. In particular, we give the definition of the bridge vertex of a G-parking function and obtain an expression of the Tutte polynomial TG(x, y) of G in terms of G-parking functions. We...

We show that if G is a simple outerplanar graph and H is a graph with the same Tutte polynomial as G, then H is also outerplanar. Examples show that the condition of G being simple cannot be omitted.