19 research outputs found
Hemisystems of small flock generalized quadrangles
In this paper, we describe a complete computer classification of the
hemisystems in the two known flock generalized quadrangles of order
and give numerous further examples of hemisystems in all the known flock
generalized quadrangles of order for . By analysing the
computational data, we identify two possible new infinite families of
hemisystems in the classical generalized quadrangle .Comment: slight revisions made following referee's reports, and included raw
dat
Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems
Inspired by some intriguing examples, we study uniform association schemes
and uniform coherent configurations, including cometric Q-antipodal association
schemes. After a review of imprimitivity, we show that an imprimitive
association scheme is uniform if and only if it is dismantlable, and we cast
these schemes in the broader context of certain --- uniform --- coherent
configurations. We also give a third characterization of uniform schemes in
terms of the Krein parameters, and derive information on the primitive
idempotents of such a scheme. In the second half of the paper, we apply these
results to cometric association schemes. We show that each such scheme is
uniform if and only if it is Q-antipodal, and derive results on the parameters
of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We
revisit the correspondence between uniform indecomposable three-class schemes
and linked systems of symmetric designs, and show that these are cometric
Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class
schemes in terms of only a few parameters, and show that any strongly regular
graph with a ("non-exceptional") strongly regular decomposition gives rise to
such a scheme. Hemisystems in generalized quadrangles provide interesting
examples of such decompositions. We finish with a short discussion of
five-class schemes as well as a list of all feasible parameter sets for
cometric Q-antipodal four-class schemes with at most six fibres and fibre size
at most 2000, and describe the known examples. Most of these examples are
related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions,
April 201
Intriguing sets of vertices of regular graphs
Intriguing and tight sets of vertices of point-line geometries have recently been studied in the literature. In this paper, we indicate a more general framework for dealing with these notions. Indeed, we show that some of the results obtained earlier can be generalized to larger classes of graphs. We also give some connections and relations with other notions and results from algebraic graph theory. One of the main tools in our study will be the Bose-Mesner algebra associated with the graph
Uniformity in Association schemes and Coherent Configurations: Cometric Q-Antipodal Schemes and Linked Systems
2010 Mathematics Subject Classification. Primary 05E30, Secondary 05B25, 05C50, 51E12
Incidence geometry from an algebraic graph theory point of view
The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense.
The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte.
Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them.
Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight.
A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5.
Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems.
The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe