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Incidence geometry from an algebraic graph theory point of view

Frédéric Vanhove (2011)
abstract
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.
Please use this url to cite or link to this publication:
author
promoter
and UGent
organization
year
type
dissertation
publication status
published
subject
keyword
finite geometry, distance-regular graphs, association schemes
pages
XIV, 243 pages
publisher
Ghent University. Faculty of Sciences
place of publication
Ghent, Belgium
defense location
Gent : Campus Sterre (S25, auditorium Emmy Noether)
defense date
2011-04-07 17:00
language
English
UGent publication?
yes
classification
D1
copyright statement
I have transferred the copyright for this publication to the publisher
id
1209078
handle
http://hdl.handle.net/1854/LU-1209078
date created
2011-04-14 15:32:33
date last changed
2017-04-28 12:11:17
@phdthesis{1209078,
  abstract     = {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 \ensuremath{-} 1, q\^{ }2) with d odd, the bound is new and even tight.
A variant of the famous Erd\unmatched{0151}s-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 \ensuremath{-} 1, q\^{ }2) for odd rank d \ensuremath{\geq} 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.},
  author       = {Vanhove, Fr{\'e}d{\'e}ric},
  keyword      = {finite geometry,distance-regular graphs,association schemes},
  language     = {eng},
  pages        = {XIV, 243},
  publisher    = {Ghent University. Faculty of Sciences},
  school       = {Ghent University},
  title        = {Incidence geometry from an algebraic graph theory point of view},
  year         = {2011},
}

Chicago
Vanhove, Frédéric. 2011. “Incidence Geometry from an Algebraic Graph Theory Point of View”. Ghent, Belgium: Ghent University. Faculty of Sciences.
APA
Vanhove, F. (2011). Incidence geometry from an algebraic graph theory point of view. Ghent University. Faculty of Sciences, Ghent, Belgium.
Vancouver
1.
Vanhove F. Incidence geometry from an algebraic graph theory point of view. [Ghent, Belgium]: Ghent University. Faculty of Sciences; 2011.
MLA
Vanhove, Frédéric. “Incidence Geometry from an Algebraic Graph Theory Point of View.” 2011 : n. pag. Print.