10:00 to 11:00 N Biggs (London School of Economics)Complex roots of chromatic polynomials I shall begin by explaining how the theory of representations of the symmetric group can be applied to the transfer matrix. This leads to explicit formulae for the chromatic polynomials of families of graphs, in which the terms correspond to partitions of positive integers. The formulae are well-suited to the application of the Beraha-Kahane-Weiss theorem, describing the limit points of zeros of the polynomials. In simple cases the individual terms can be written explicitly as powers of polynomials, and the resulting limit curves are (parts of) closed curves. In the general case the curves can have end-points and singularities, and I shall discuss some of the interesting phenomena that can occur. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 J Jacobsen ([Paris-Sud])Representations and partition function zeros of the Potts model with and without boundaries We consider the following four classes of models defined on the annulus, listed in order of increasing generality: 1. Restricted height models with face interactions 2. Fully-packed loop models based on the Temperley-Lieb algebra 3. TL loop models with one boundary generator 4. TL loop models with two boundary generators All of these are different formulations of the Potts model (Tutte polynomial), where, for classes 3 and 4, spins on the rims of the annulus are required to take a different number of states than bulk spins. We show how each class reduces to the one preceding it, provided that one of its parameters takes particular, magical values. In an appropriate part of the phase diagram, these reductions lead to partition function zeros in the more general model. In particular, the reduction from class 2 to 1 gives zeros of the chromatic polynomial at the Beraha numbers, provided one is inside the so-called Berker-Kadanoff phase. INI 1 12:30 to 13:45 Lunch at Churchill College 14:00 to 14:30 Dominant traits in the zeros of two-variate two-terminal reliability polynomials Various polynomials appear in graph theory: chromatic polynomials, flow polynomials, all-terminal reliability polynomials. All these polynomials are special instances of the Tutte polynomial, and the location of their complex zeros has been actively studied (Biggs, Brown & Colbourn, Chang & Shrock, Royle, Sokal, Welsh, to name but a few). General results have been established for the zeros of recursively defined families of polynomials (Beraha, Kahane, and Weiss): the zeros aggregate asymptotically to sets of algebraic curves and sometimes to isolated points. In this contribution, we address the two-terminal (or end-to-end) reliability polynomial Rel2, which corresponds to the connectivity of two sites of a graph. Admittedly, this is no graph invariant. However, instead of using the variable $p$ for the (uniform) probability for correct operation of the edges of the graph, we may now consider two variables $p$ and $\rho$ for the edge and site reliabilities, respectively. For a given graph and a pair of sites, we can study the location of the complex zeros of Rel2($p$,$\rho$) as a function of $\rho$. We present results on a few recursive families of graphs for which the two-terminal reliability polynomial may be computed exactly as a function of $p$,$\rho$ and $n$ (the size of the graph); the associated generating functions have also been derived. The asymptotic (for $n$ going to infinity) location of zeros has been investigated and exhibits: - the usual sets of algebraic curves, - appearing (or disappearing) sets of isolated zeros in some cases, - an expansion of the general structure as $\rho$ goes to 0, which can be described by a power law. The set of zeros also undergoes several structural transitions at specific, critical values $\rho_c$ --- which are algebraic --- as $\rho$ decreases from 1 to 0. More surprisingly, by slightly changing the building blocks of the recursive graphs, we may also numerically observe: - particular values of $\rho$ for which the sets of zeros look much smoother, - existence (or not!) of several prevailing sets of isolated zeros with different symmetries (of orders 3 and 13, for instance), - different exploding'' families of algebraic curves with different symmetries and expansion rates, too. Asymptotic, analytical expressions for the above phenomena can actually be deduced from the generating function. INI 1 14:30 to 15:00 Integer symmetric matrices with spectral radius at most 2.019 All graphs with spectral radius (of their adjacency matrix) at most sqrt(2+sqrt(5)) = 2.05817... are known. In joint work with James McKee, we try to extend this result to general symmetric matrices with integer entries. As is clear from the title, we do not get quite as far. Apart from some trivial examples, all the matrices described by the title are represented by 'charged signed graphs' having adjacency matrices with entries 0,1 or -1. INI 1 15:00 to 15:30 Tea 15:30 to 16:30 Problem session INI 1 18:00 to 19:00 Dinner at Churchill College