Gordon F. Royle

Flocks and ovals

An infinite family of q-clans, called the Subiaco q-clans, is constructed for q=2e. Associated with these q-clans are flocks of quadratic cones, elation generalized quadrangles of order (q2, q), ovals of PG(2, q) and translation planes of order q2 with kernel GF(q). It is also shown that a q-clan, for q=2e, is equivalent to a certain configuration of q+1 ovals of PG(2, q), called a herd.

Matroids with nine elements

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.

Sets of type ( m, n ) in the affine and projective planes of order nine

In this paper we give the results of exhaustive computer searches for sets of points of type (m, n) in the projective and affine planes of order nine. In particular, as the list of planes of order nine is known to be complete, our results are also complete. We also examine all known constructions of sets of type (m, n) that apply to the planes of order nine, in an attempt to summarise and extend all existing knowledge about such sets. The contrast between known constructions and our computer results leads us to conclude that sets of type (m, n) are far more numerous than was previously thought.

The Brown--Colbourn conjecture on zeros of reliability polynomials is false

We give counterexamples to the Brown-Colbourn conjecture on reliability polynomials, in both its univariate and multivariate forms. The multivariate Brown Colbourn conjecture is false already for the complete graph K4. The univariate Brown-Colbourn conjecture is false for certain simple planar graphs obtained from K4 by parallel and series expansion of edges. We show, in fact, that a graph has the multivariate Brown Colbourn property if and only if it is series parallel.

Computing Tutte Polynomials

The Tutte polynomial of a graph, also known as the partition function of the q-state Potts model is a 2-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chromatic polynomial and flow polynomial as partial evaluations, and various numerical invariants such as the number of spanning trees as complete evaluations. However despite its ubiquity, there are no widely available effective computational tools able to compute the Tutte polynomial of a general graph of reasonable size. In this article we describe the implementation of a program that exploits isomorphisms in the computation tree to extend the range of graphs for which it is feasible to compute their Tutte polynomials, and we demonstrate the utility of the program by finding counterexamples to a conjecture of Welsh on the location of the real flow roots of a graph.

Java as a teaching language—opportunities, pitfalls and solutions

This paper describeqeriences in teaching Java to Computer Science students in two Australian universities. The paper highlights some of the problems encountered in teaching Java, and some of the areas that needed careful treatment. Based on these experiences -we suggest an unorthodox ‘objects-only” approach to introducing Java to new students.

Classification of ovoids inPG(3, 32)

A classification of the ovoids inPG(3, 32) is completed with the aid of a computer. The ovoids are examined in terms of which ovals can possibly appear as secant plane sections. A weak necessary condition for two ovals to appear together as plane sections of an ovoid surprisingly turns out to be sufficient to demonstrate that the only possible secant plane sections are translation ovals. A known result regarding ovoids with such plane sections then identifies the ovoids as either elliptic quadrics or Tits ovoids.

An orderly algorithm and some applications in finite geometry

Abstract An algorithm for generating combinatorial structures is said to be an orderly algorithm if it produces precisely one representative of each isomorphism class. In this paper we describe a way to construct an orderly algorithm that is suitable for several common searching tasks in combinatorics. We illustrate this with examples of searches in finite geometry, and an extended application where we classify all the maximal partial flocks of the hyperbolic and elliptic quadrics in PG(3, q ) for q ⩽ 13.

Hyperovals in the Known Projective Planes of Order 16

We construct by computer all of the hyperovals in the 22 known projective planes of order 16. Our most interesting result is that four of the planes contain no hyperovals, thus providing counterexamples to the old conjecture that every finite projective plane contains an oval.

Symmetric squares of graphs

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.