The tensor product (G1, G2) of a graph G1 and a pointed graph G2 (containing one distinguished edge) is obtained by identifying each edge of G1 with the distinguished edge of a separate copy of G2, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to colored gra...

We introduce a multiplicity Tutte polynomial M(x, y), which generalizes the ordinary one and has applications to zonotopes and toric arrangements. We prove that M(x, y) satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x, y), li...

There are many polynomials which can be associated with a graph G, the most well known perhaps being the Tutte polynomial T (G; x, y) (cf. [B74] or [T54]). In particular, for specific values of x and y, T (G; x, y) enumerates various features of G. For example, T (G; 1, 1) is just the number of spanning trees of G, T (G; 2, 0) is the number of acyclic orientations of G, T (G; 1, 2) is the numbe...

We consider linear codes in the metric space with the Niederreiter-Rosenbloom-Tsfasman (NRT) metric, calling them linear ordered codes. In the first part of the paper we examine a linear-algebraic perspective of linear ordered codes, focusing on the distribution of “shapes” of codevectors. We define a multivariate Tutte polynomial of the linear code and prove a duality relation for the Tutte po...

Jones polynomials and Kauuman polynomials are the most prominent invariants of knot theory. For alternating links, they are easily computable from the Tutte polynomials by a result of Thistlethwaite (1988), but in general one needs colored Tutte polynomials, as introduced by Bollobas and Riordan (1999). Knots and links can be presented as labeled planar graphs. The tree width of a link L is dee...