Knot invariants and the Bollobas-Riordan polynomial of embedded graphs

For a graph G embedded in an orientable surface \Sigma, we consider associated links L(G) in the thickened surface \Sigma \times I. We relate the HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We use knot theory to prove results about graph polynomials and, after discussing questions of equivalence of the polynomials, we go on to use our formulae to prove a duality relation for the Bollobas-Riordan polynomial. We then consider the specialization to the Jones polynomial and recent results of Chmutov and Pak to relate the Bollobas-Riordan polynomials of an embedded graph and its tensor product with a cycle.Comment: v2: minor corrections, to appear in European Journal of Combinatoric

The Go polynomials of a graph

Abstract This paper introduces graph polynomials based on a concept from the game of Go. Suppose that, for each vertex of a graph, we either leave it uncoloured or choose a colour uniformly at random from a set of available colours, with the choices for the vertices being independent and identically distributed. We ask for the probability that the resulting partial assignment of colours has the following property: for every colour class, each component of the subgraph it induces has a vertex that is adjacent to an uncoloured vertex. In Go terms, we are requiring that every group is uncaptured. This deÿnition leads to Go polynomials for a graph. Although these polynomials are based on properties that are less "local" in nature than those used to deÿne more traditional graph polynomials such as the chromatic polynomial, we show that they satisfy recursive relations based on local modiÿcations similar in spirit to the deletion-contraction relation for the chromatic polynomial. We then show that they are #P-hard to compute in general, using a result on linear forms in logarithms from transcendental number theory. We also brie y record some correlation inequalities

Eigenvalue interlacing and weight parameters of graphs

Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some "weights" (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights "regularize" the graph, and hence allow us to define a kind of regular partition, called "pseudo-regular," intended for general graphs. Here we show how to use interlacing for proving results about some weight parameters and pseudo-regular partitions of a graph. For instance, generalizing a well-known result of Lov\'asz, it is shown that the weight Shannon capacity $\Theta^*$ of a connected graph \G, with $n$ vertices and (adjacency matrix) eigenvalues $\lambda_1>\lambda_2\ge\...\ge \lambda_n$, satisfies \Theta\le \Theta^* \le \frac{\|\vecnu\|^2}{1-\frac{\lambda_1}{\lambda_n}} where $\Theta$ is the (standard) Shannon capacity and \vecnu is the positive eigenvector normalized to have smallest entry 1. In the special case of regular graphs, the results obtained have some interesting corollaries, such as an upper bound for some of the multiplicities of the eigenvalues of a distance-regular graph. Finally, some results involving the Laplacian spectrum are derived. spectrum are derived

Computing Node Polynomials for Plane Curves

According to the G\"ottsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. These "node polynomials" N_delta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14. Furthermore, we improve the threshold of polynomiality and verify G\"ottsche's conjecture on the optimal threshold up to delta <= 14. We also determine the first 9 coefficients of N_delta(d), for general delta, settling and extending a 1994 conjecture of Di Francesco and Itzykson.Comment: 23 pages; to appear in Mathematical Research Letter