Xavier Hubaut

Strongly regular graphs

In this paper we have tried to summarize the known results on strongly regular graphs. Both groupal and combinatorial aspects of the theory have been included. We give the list of all known strongly regular graphs and a large bibliography of this subject.

Sets of even type in PG(3, 4), alias the binary (85, 24) projective geometry code

Abstract To characterize Hermitian varieties in projective space PG(d, q) of d dimensions over the Galois field GF(q), it is necessary to find those subsets K for which there exists a fixed integer n satisfying (i) 3 ⩽ n ⩽ q − 1, (ii) every line meets K in 1, n or q + 1 points. K is called singular or non-singular as there does or does not exist a point P for which every line through P meets K in 1 or q + 1 points. For q odd, a non-singular K is a non-singular Hermitian variety (M. Tallini Scafati “Caratterizzazione grafica delle forme hermitiane di un Sr, q” Rend. Mat. Appl. 26 (1967), 273–303). For q even, q > 4 and d = 3, a non-singular K is a Hermitian surface or “looks like” the projection of a non-singular quadric in PG(4, q) (J.W.P. Hirschfeld and J.A. Thas “Sets of type (1, n, q + 1) in PG(d, q)” to appear). The case q = 4 is quite exceptional, since the complements of these sets K form a projective geometry code, a (21, 11) code for d = 2 and an (85, 24) code for d = 3. The full list of these sets is given.

A Class of Strongly Regular Graphs Related to Orthogonal Groups

Publisher Summary This chapter describes a class of regular graphs related to PO 2n+l (q). If q is odd, the vertices of the graphs may be seen as a set of points off a quadric in PG(2n,q). It has been noticed that regular graphs of the same kind occur in symplectic spaces over GF(2 r ). The chapter presents a unified description of those graphs for a field or arbitrary characteristic.

The Problem of Representation Based Upon two Criteria

Abstract The problem considered here is that of electing a delegation when the population of electors is separated into categories, the division being based uoon one or several criteria. A two criteria-based situation currently exists in the Netherlands. Using a slightly more general version of this system, we proved that the only tyne of situation avoiding imDossibi lities is one where each criterion only presents two alternatives. Furthermore, setting a number of conditions, as weak as possible, which must be respected in order to ensure fair representation, the same result remains - which is not so surprising as it could have been suggested by Arrows famous theorem.