No abstract

We show that if G is a group of automorphisms of a thick finite generalised quadrangle Q acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.

We consider the location of zeros of four related classes of polynomials, one of which is the class of chromatic polynomials of graphs. All of these polynomials are generating functions of combinatorial interest. Extensive calculations indicate that these polynomials often have only real zeros, and we give a variety of theoretical results which begin to explain this phenomenon. In the course of the investigation we prove a number of interesting combinatorial identities and also give some new sufficient conditions for a polynomial to have only real zeros.

We consider symmetric powers of a graph. In particular, we show that the spectra of the symmetric square of strongly regular graphs with the same parameters are equal. We also provide some bounds on the spectra of the symmetric squares of more general graphs. The connection with generic exchange Hamiltonians in quantum mechanics is discussed in Appendix A.

We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this catalogue. Our computation confirms and extends the results obtained in the 1960s by Blackburn, Crapo and Higgs. The matroids and associated data are stored in an on-line database, and we give three short examples of the use of this database.