For Fp the field on prime number p > 3 elements, it has been conjectured that there are just p+3 evaluations of the Tutte polynomial in Fp which are computable in polynomial time. In this note it is shown that if p 6≡ −1mod 12 then there are further polynomial-time computable evaluations. 1 Definitions and introduction Let G = (V,E) denote a graph, with loops and parallel edges permitted, and G...

One of the most important numerical quantities that can be computed from a graph G is the two-variable Tutte polynomial. Specializations of the Tutte polynomial count various objects associated with G, e.g., subgraphs, spanning trees, acyclic orientations, inversions and parking functions. We show that by partitioning certain simplicial complexes related to G into intervals, one can provide com...

We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) ZG(q,w) of a graph G with general complex edge weights w = {we}. This generalizes a result of Sokal [22] that applied only within the complex antiferromagnetic regime |1+we| ≤ 1. Our proof uses the polymer-gas representatio...

We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) ZG(q,w) of a graph G with general complex edge weights w = {we}. This generalizes a result of Sokal [22] that applied only within the complex antiferromagnetic regime |1+we| ≤ 1. Our proof uses the polymer-gas representatio...

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.