 Author
 Frédéric Vanhove (UGent)
 Promoter
 Frank De Clerck (UGent) and John Bamberg (UGent)
 Organization
 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 distanceregular 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 msystems 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 ErdősKoRado problem is considered in Chapter 5, where we study sets of pairwise nontrivially 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.
 Keywords
 finite geometry, distanceregular graphs, association schemes
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU1209078
 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.
@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 distanceregular 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 msystems 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}sKoRado problem is considered in Chapter 5, where we study sets of pairwise nontrivially 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,distanceregular 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}, }