關於射影、仿射兩型關聯結構之研究

Abstract

本論文討論下列四族距離正則圖形及其衍生之關聯結構(incidence structures)：Grassmann圖，雙線性型圖(bilinear forms graphs)，半 對偶極圖(half dual polar graphs)，交錯型圖(alternating forms graphs)。由於其極大點團(maximal cliques)具有射影空間(projective spaces)或仿射空間(affine spaces)的結構，上述四族圖形及其關聯結 I. 利用A. Neumaiern所提出之古典參數(classical parameters)整合 Grassmann圖Jq(n,d)與雙線性型圖Hq(n,d)於同一模式基礎上予以刻劃。 II. 除探討半對偶極圖Dn,n(q)與交錯型圖Alt(n,q)之幾何性質外，特別 針對 n=4 (即直徑為 2)的情形予以一致性之刻劃。 III. 我們提出類半 對稱設計(Quasi semisymmetric designs)之概念以描述上述四族圖形的 幾何結構，並引進下列幾何條件以探討參數之相關性質： (＊)-條件： 若B1,B2,B3為含有兩個以上共同交點之相異區組，則B1,B2,B3含有μ個共 同交點。 (△)-條件： 任三個兩兩相連的點至少包含在一個共同區組內 。 Cameron & Drake 已完成參數μ＝λ的情形。我們考慮參數μ＝λ－1 的情形，並刻劃滿足(＊)和(△)條件的設計。 In this thesis, we study the following four classes of distance- regular graphs and their associated incidence structures: Grassmann graphs J_q(n,d), bilinear forms graphs H_q(n,d), half dual polar graphs D_{n,n}(q), and alternating forms graphs Alt( n,q). The associated incidence structures of the distance- regular graphs mentioned above are called of projective type or affine type depending on their maximal cliques being projective spaces or affine spaces. The main results of this thesis are briefly described as follows: I. We give a unified characterization of the Grassmann graphs J_q(n,d) and the bilinear forms graphs H_q(n,d) as the distance-regular graphs with classical parameters (in the sence of Neumaier) and some extra conditions. II. In addition to the geometric properties of the half dual polar graphs D_{n,n}(q) and the alternating forms graphs, Alt(n,q), we give a unified characterization of D_{4,4}(q) and Alt(4,q) as the strongly regular graphs with classical parameters and some extra conditions. III. We propose the notion of quasi semisymmetric designs (QSSD) as a framework. In particular, we impose the following two extremal conditions on QSSD with nexus and parameters (v,k, [.lambda.],[. mu.]): (*)-condition: if B1, B2, B3 are three distinct blocks with |B1 .intersetion. B2 .intersection. B3| .gtoreq. 2, then then |B1. intersection. B2. intersection. B3| .gtoreq. .mu. (.DELTA.)-condition: any three distinct pairwise collinear points are in at least one common block. Two classes of QSSD with nexus, i.e., QSSD(v,k,[.lambda.], [.mu.]) with .mu.=. lambda. and .mu.=.lambda.-1, satisfying the above extremal conditions are classified.