在一些集合系上的Fisher性質以及相交性質的探討

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(1)國立交通大學應用數學系碩士論文. 在一些集合系上的 Fisher 性質以及相交性質的探討 A Study of Fisher Type Property and Intersecting Properties over Some Set Systems. 研究生：蔡嘉文指導老師：黃大原. 教授. 中華民國九十三年六月.

(2) 在一些集合系上的 Fisher 性質以及相交性質的探討 A Study of Fisher Type Property and Intersecting Properties over Some Set Systems. 研究生：蔡嘉文. Student: Chia-Wen Tsai. 指導老師：黃大原教授. Advisor: Dr. Tayuan Huang. 國立交通大學應用數學系碩. 士. 論. 文. A Thesis Submitted to Department of Applied Mathematics College of Science National Chiao Tung University In partial Fulfillment of Requirement For the Degree of Master In Applied Mathematics June 2004 Hsinchu, Taiwan, Republic of China. 中華民國九十三年六月.

(3) 在一些集合系上的 Fisher 性質以及相交性質的探討研究生：蔡嘉文. 指導老師：黃大原. 教授. 國立交通大學應用數學系摘. 要. 本論文首先針對在各種區組設計上的 Fisher 不等式進行研究，如 t區組設計、可分解設計、以及 2s-區組設計，這些不等式利用關聯矩陣為出發點，以線性代數為工具予以證明。其次討論一些區組設計，如 2s區組設計，3-(10, 4, 1)設計，以及 3-(2k, 4, 1)設計。我們也以區組設計的性質、強正則圖、以及正交陣列為例，來說明如何利用二次函數的技巧來推導出一些組合性質。. 中華民國九十三年六月 i.

(4) A Study of Fisher Type Property and Intersecting Properties over Some Set Systems Student : Chia-Wen Tsai. Advisor : Dr. Tayuan Huang. Department of Applied Mathematics National Chiao Tung University Hsinchu 300, Taiwan, R.O.C.. Abstract Fisher’s type inequalities over various designs such as t-designs, resolvable designs, and 2s-designs are studied in this thesis. The unified technique in terms of their incidence matrices, within the framework of linear algebra is emphasized. Moreover, the intersections among pairs of blocks of some designs such as 2s-design, 3-(10, 4, 1) design and 3-(2k, 4, 1) designs, are also studied, the technique of using quadratic functions is also emphasized.. ii.

(5) A Study of Fisher Type Property and Intersecting Properties over Some Set Systems Student: Chia-Wen Tsai. Advisor: Tayuan Huang. Department of Applied Mathematics National Chiao Tung University Hsinchu 30050 Taiwan R.O.C. June 17, 2004.

(6) Contents 1 Introduction. 1. 2 Preliminaries 2.1 Set Systems and Their Incidence Matrices . . . . . . . . . . . . . . . 2.2 Combinatorial t-designs . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 2 2. 3 Some Techniques in terms of Matrices and Discriminants 4 3.1 Fisher Inequalities over Various Designs . . . . . . . . . . . . . . . . . 4 3.2 The Intersections among Pairs of Blocks of Some Designs . . . . . . . 6 3.3 Combinatorial Information derived from Quadratic functions . . . . . 10 4 Quasi-symmetric designs and Strongly Regular 4.1 Quasi-symmetric designs with Good Blocks . . . 4.2 A Class of Strongly Regular Designs SRD . . . 4.2.1 Quasi-semi symmetric designs . . . . . . 4.2.2 Strongly Regular Designs . . . . . . . . . 4.2.3 Some properties of SRD(v, k, λ, µ) . . . Reference. designs . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 14 14 18 18 19 20 23. ii.

(7) 1. Introduction. Let X be a finite set, and Φ ⊆ 2X , then (X, Φ) is called a set system. The notion of set systems usually provides a convenient model for combinatorial structures. Each set system can be expressed in terms of its incidence matrix M . More precisely, an incidence matrix M of a set system Π = (X, Φ) is row-indexed by elements of X and column-indexed by members of Φ respectively such that the entry of M along the x row and F th column is 1 if and only if x ∈ F and 0 otherwise. Many results and techniques in linear algebra can be used to derive some information for combinatorial structure under consideration through its incidence matrix. In Chapter 2, we introduce some kinds of set systems. Strongly regular graph is a set system with X = V (Γ), Φ = {Γ1 (u) : for all u ∈ V (Γ)} and some properties from adjacency of vertices. The family of t-designs forms another family of set system. Symmetric designs and quasi-symmetric designs are two classes of 2designs whose incidence matrices can be associated with strongly regular graphs. Fisher’s inequality over various designs, such as t-designs, resolvable designs and 2s-designs, will be presented in Section 3.1 and proved by analyzing the incidence matrices. Besides, Fisher’s inequality also can be stated in terms of the terminology of graphs. In Section 3.2, the intersections among pairs of blocks of some designs including 2s-design, 3-(10, 4, 1) design and 3-(2k, 4, 1) designs are discussed, and graphs which has friendship property are introduced. The technique using quadratic functions is introduced in Section 3.3, its applications will be illustrated in terms of a few examples including some properties of designs, strongly regular graphs and orthogonal arrays respectively. Quasi-symmetric designs are those 2-designs with exactly two different intersection numbers among each pair of their blocks, and hence their block graphs are strongly regular. Since the examples of quasi-symmetric designs which are not symmetric designs, affine designs or linear spaces are rather rare. An approach in the study of quasi-symmetric designs with some disjoint blocks is to consider their substructures and induced substructures. In Section 4.1, we consider those quasi-symmetric designs having good blocks.. 1.

(8) 2. Preliminaries. 2.1. Set Systems and Their Incidence Matrices. An incidence structure Π = (X, B) consists of a finite set X, together with a collection B of subsets of X. The elements in X are called points, and the subsets in B are called blocks, we call a point x lies in a block B if x ∈ B. An incidence structure Π = (X, B) is usually denoted by its incidence matrix M in the following sense that M is row-indexed (respective column-indexed) by points (respective by blocks) such that the entry M (x, B) of M at the row indexed by the points x and the column indexed by the block B is 1 if x ∈ B and 0 otherwise. A residue structure associated with a point a is defined as follows. Let N (a) = {a} ∪ {x ∈ X|x is collinear with a}. If B is a block not containing a, the residue B a of B with respect to a is defined to be B − N (a). Let B a = {B a |B ∈ B with a ∈ B}. Then the residue of Π = (X, B) with respect to a is defined to be Πa = (X − N (a), B a ). Definition 1. Let Γ be a k-regular graph of order v with the following properties that each pair of adjacent vertices has λ common neighbors and each pair of nonadjacent vertices has µ common neighbors. Then Γ is said to be a strongly regular graph with parameters (v, k, λ, µ). 2.2. Combinatorial t-designs. Definition 2. Let v, k and λ be positive integers such that v > k ≥ 2. A (v, k, λ)balanced incomplete block design (which we abbreviate to (v, k, λ)-BIBD), which is also called a 2-(v, k, λ) design, is a pair (X, B) such that the following properties are satisfied: 1. X is a set of v elements called points, 2. B is a collection of subsets of X called blocks, 3. each block contains exactly k points, and 4. every pair of distinct points is contained in exactly λ blocks. Let X = {x1 , x2 , ..., xv } and B = {B1 , B2 , ..., Bb }. The incidence matrix of (X, ( B) is the (0, 1)-matrix M = (mij ) of order v × b defined by the rule mij = 1 if xi ∈ Bj 0 if xi 6∈ Bj A 2-(v, k, λ) design Π = (X, B) with b = v is called a symmetric design. It is known that the followings are equivalent. 2.

(9) 1. |Bi ∩ Bj | = λ for any pair Bi , Bj ∈ B, 2. Π is a symmetric design. A quasi-symmetric design is a 2-design with the number of points in the intersection of two blocks takes only two values. Let x < y be the two cardinalities of block intersection in the quasi-symmetric design. The block graph of a quasi-symmetric design has as vertices the blocks, two vertices adjacent if they intersect y points. Definition 3. A t-design with parameters (v, k, λ) (or a t-(v, k, λ) design) is a pair Π = (X, B), where X is a set of ’points’ of cardinality v, and B a collection of kelement subsets of X called ’blocks’, with the property that any t points are contained in precisely λ blocks.. 3.

(10) 3. Some Techniques in terms of Matrices and Discriminants. Some combinatorial informations can be derived from matrices and quadratic functions as well. For example, the fact that the rank of a matrix is always less than or equal to its numbers of rows and of columns leads to an inequality. Among many others, Fisher inequality is such an example. In this section, we will explore such a technique for Fisher inequalities for BIBD and their variations. Moreover, a quadratic function can also be associated with some combinatorial problems, and its discriminant usually provides another inequality with combinatorial interests. We will explain this technique in Section 3.2 in terms of a few typical examples. 3.1. Fisher Inequalities over Various Designs. Some combinatorial informations can be derived from matrices and quadratic functions as well. For example the fact that the rank of a matrix is always less than or equal to its numbers of rows and of columns leads to an inequality. Among many others, Fisher inequality is such an example. In this section, we will explore such a technique for Fisher inequalities for BIBD and their variations. Theorem 3.1.1 (Fisher’s Inequality). For a t-(v, k, λ) design Π = (X, B), then 1. b ≥ v if t = 2; 2. b ≥ v + r − 1 if t = 2, and it is resolvable with r parallel classes; the equality 2 holds with r = k + λ; any two blocks from different classes meet at kv points. ¡¢ 3. b ≥ vs if t = 2s, and k ≤ v − s. Proof. Let Π = (X, B) be a 2-(v, k, λ) design, where X = {x1 , . . . , xv } and B = {B1 , . . . , Bb }. Let M be the incidence matrix of this 2-design, and define sj to be the jth column of M , note that s1 , . . . , sb all are v-dimensional vectors in the real vector space Rv . Let b X S={ αj sj : α1 , . . . , αb ∈ R} j=1 v. then S is the subspace of R spanned by S = {sj : 1 ≤ j ≤ b}, it consists of all linear combinations of the vectors s1 , . . . , sb , it suffices to show that ei ∈ S for 1 ≤ i ≤ v. P First, we observe that bj=1 sj = (r, . . . , r)T and then  1 1  .  sj =  ..  . r 1 . b X j=1. 4.

(11) Next, fix a value i, 1 ≤ i ≤ v. Then we have  λ  .  sj = (r − λ)ei +  ..  . X {j:xi ∈Bj }. λ. Combine them to obtain ei =. X {j:xi ∈Bj }. b. X λ 1 sj , sj − r(r − λ) r−λ j=1. a linear combination of s1 , . . . , sb , as required. + 1 and ni = ivk for 1 ≤ i ≤ r. Suppose that the blocks To prove 2, let mi = (i−1)v k are labelled with the r parallel classes Πi = {Bj : mi ≤ j ≤ ni }, 1 ≤ i ≤ r, so we P i P 1 P i have that nj=m s = (1, . . . , 1)T , for 1 ≤ i ≤ r, and smi = nj=m s − nj=m s 1 j i j i +1 j 0 for 2 ≤ i ≤ r. In other words, the r − 1 vectors in the set S = {sm2 , . . . , smr } can be expressed as linear combinations of the b − r + 1 vectors in S − S 0 . Now, since the b vectors in S span Rv , it follows that the b − r + 1 vectors in S − S 0 span Rv . Since Rv has dimension v and is spanned by a set of b − r + 1 vectors, it must be the case that b ≥ v + r − 1. To prove 3, we use a modified incidence matrix Ms , whose columns are indexed by blocks and rows by s-sets of points, with (S, B) entry 1 if S ⊂ B, 0 otherwise, ¡¢ then Ms is a vs × b matrix, and it suffices to show that the columns of Ms span v R(s) . Accordingly, let c be the column of M with label B, and let e be the vector B. s. s. with 1 in the position labelled S and 0 in all other positions (the unit basis vector P P corresponding to S). For xj = |B∩S|=i rB , |S 0 ∩S|=j eS 0 , 0 ≤ j ≤ k, and yi = 0 ≤ i ≤ s. Because ¶ i i µ X X X X X X s−j yi = rB = v2s−j,s+i−j ( eS 0 ) eS 0 = i − j 0 0 0 j=0 j=0 |B∩S|=i. |S ∩S|=j. |S∩S |=j. B ⊇S |B ∩ S| = i. where vp,q denotes the number of blocks intersecting a given p-set P in a given qsubset Q. Then we have a system of s + 1 linear equations for the xj in terms of the yi . The coefficient matrix is triangular, and its diagonal entries v2s−i,s are non-zero (since s ≤ k ≤ v − s). So the equations have a unique solution. In particular, xs = eS is a linear combination of the yi , and so is in the row space of Ms . Fisher Inequality can also be stated in terms of the terminology of graphs. Theorem 3.1.2. Let G1 , . . . , Gb be (not necessarily distinct) complete subgraphs of Kv , v ≥ 2, each of order at most n − 1 such that every edge of Kv belongs to the same number λ ≥ 1 of Gi s. Then b ≥ v. 5.

(12) Proof. Let V (Kv ) = {a1 , . . . , an }, and let M be the v × b incidence matrix of the cover Kv = ∪bi=1 Gi , i.e., Mij = 1 if ai ∈ Gj ; and Mij = 0 otherwise. Then   m1 λ · · · λ  ..   λ m λ .  2 , MMT =   ..  .. . λ   . λ λ · · · λ mv where mi is the number of Gj , 1 ≤ j ≤ b, containing ai . Since each edge of Kv belongs to λ of Gi s, each point of Kv belongs to at least λ of Gi s. If there exists a point ak of Kv belongs to exactly λ of Gi s, then {a1 , . . . , an } − {ak } belong to these λ of Gi s. (Because each of the edges that connects ak and {a1 , . . . , an } − {ak } belongs to λ of Gi s) thus, each of these λ of Gi s has order v, a contradiction. So every point belongs to more than λ of Gi s. Since every point of Kv belongs to more than λ of Gi s, let mi = λ + ki with ki ≥ 1 for 1 ≤ i ≤ v. Then ¯ ¯ ¯ ¯ ¯ λ + k1 ¯ ¯ λ + k λ ... λ ¯ λ · · · λ 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ .. ¯ λ ¯ ¯ −k1 k2 0 0 ¯ λ + k2 λ . T ¯ ¯ ¯ ¯ det(M M ) = ¯ .. .. .. ¯=¯ .. . 0 ¯¯ . 0 . ¯ . λ λ ¯ ¯ ¯ ¯ ¯ ¯ ¯ λ 0 . . . kv ¯ ··· λ λ + kv ¯ ¯ −k1 ¯ ¯ ¯ λ + k + λ k1 + . . . + λ k1 λ . . . λ ¯ 1 ¯ ¯ kn k2 ¯ ¯ v v X ¯ 0 k2 0 0 ¯ 1 Y ¯ ¯ ) ki = (1 + λ =¯ .. .. k . 0 ¯¯ . 0 ¯ i=1 i i=1 ¯ ¯ ¯ 0 · · · 0 kv ¯ is non-zero. Thus, rank(M M T ) = v ≤ rank(M ) ≤ min{v, b}, and b ≥ v required.. 3.2. The Intersections among Pairs of Blocks of Some Designs. Theorem 3.2.1. Suppose that Π = (X, B) is a 2-(v, k, λ) design with its incidence matrix M , then the following are equivalent 1. b = v, 2. r = k, and 3. M T M = (k − β)I + βJ for some constant β. Proof. For a 2-(v, k, λ) design Π = (X, B), bk = vr and hence the first two are equivalent. 6.

(13) The matrix M T M is a b × b matrix. If k 6= β then det(M T M ) 6= 0, so rank(M T M ) = b ≤ rank(M ) ≤ min{v, b} ≤ v, hence b = v because of the Fisher inequality b ≥ v. This shows that 3. ⇒ 1. Let B = {B1 , . . . , Bb } and define sj to be the jth column of M , note s1 , . . . sb are all v-dimensional vectors in the real vector space Rv . Fix h, 1 ≤ h ≤ b, then  λ X  .  ((r − λ)ei +  .. ) . X. X. sj =. {i:xi ∈Bh } {j:xj ∈Bj }. {i:xi ∈Bh }. . . λ. λ b X λk  ..  sj , = (r − λ)sh + k  .  = (r − λ)sh + r j=1 λ and. X. X. {i:xi ∈Bh } {j:xj ∈Bj }. sj =. b X. |Bh ∩ Bj |sj . Since b = v, B = {B1 , . . . , Bb } is a basis. j=1. of Rv , the left coefficients of sj and the right coefficients of sj must equal, it follows that |Bh ∩ Bj | = λ whenever h 6= j, M T M = (k − λ)I + λJ as required, this shows that 1. ⇒ 3.. Theorem 3.2.2. Let B be a collection of k-subsets of a v-set X, where 2s ≤ k ≤ v − s. Then any two of the following conditions imply the third one: 1. (X, B) is a 2s-design; 2. the cardinality of the intersection of two distinct blocks in B takes just s distinct values; ¡¢ 3. |B| = vs . ¡¢ Proof. 1, 2 ⇒ 3: (X, B) is a 2s-design, and 2s ≤ k ≤ v − s. Then b ≥ vs and the cardinality of the intersection of two distinct blocks in B takes just s distinct values. Now, we consider its incidence matrix Ms as above. Then  ¡k ¢  1 · · · 1 s ¡k ¢  ..   1 1 .  T s   . Ms Ms =  .  ... .  . 1 1  ¡k ¢ 1 ··· 1 s b×b ¡¢ ¡¢ Because ks > 1, det(MsT Ms ) > 0. Thus, rank(MsT Ms ) = b. Then min{b, vs } = ¡¢ rank(Ms ) ≥ rank(MsT Ms ) = b, i.e., b ≤ vs = |B| 7.

(14) An example of a 3-(10, 4, 1) design is given below: {1, 2, 4, 5} {3, 4, 6, 7} {5, 6, 8, 9} {1, 7, 8, 0} {2, 3, 9, 0}. {1, 2, 3, 7} {3, 4, 5, 9} {1, 5, 6, 7} {3, 7, 8, 9} {1, 5, 9, 0}. {1, 3, 5, 8} {3, 5, 7, 0} {2, 5, 7, 9} {1, 4, 7, 9} {1, 3, 6, 9}. {2, 3, 5, 6} {4, 5, 7, 8} {6, 7, 9, 0} {1, 2, 8, 9} {1, 3, 4, 0}. {2, 3, 4, 8} {4, 5, 6, 0} {2, 6, 7, 8} {4, 8, 9, 0} {1, 2, 6, 0}. {2, 4, 6, 9} {1, 4, 6, 8} {3, 6, 8, 0} {2, 5, 8, 0} {2, 4, 7, 0}. If Y = {1, 2, 3, 5, 0}, then Y does not contain any block, and similarly, Y C does not contain any block either. The following theorem shows that can be generalized to any 3-(10, 4, 1) design. We then further show that similar condition holds in general cases. Theorem 3.2.3. If Y is a set of 5 points of a 3-(10, 4, 1) design containing no block, then the complement of Y contains no block either. Proof. For a 3-(10, 4, 1) design, it has 30 blocks, and n1 = 12, n2 = 4, n3 = 1, where ni is the number of blocks containing a fixed set of i points. Suppose there exists a set Y of 5 points containing no blocks, but Y containing a block, say B1 , without loss of generality, we may assume that Y = {1, 2, 3, 4, 5}, and the transpose of the first column of its incidence matrix is (0, 0, 0, 0, 0, 1, 1, 1, 1, 0), then Y ∩ B1 = ∅. Let D = {Bi : Y ∩ Bi 6= ø, i = 1, 2, . . . , 30}, then |D| ≤ 29 because Y ∩ B1 = ∅. On the other hand, the principal of inclusion and exclusion shows that µ ¶ µ ¶ µ ¶ 5 5 5 × 1 = 30, ×4+ × 12 − |D| = 1 2 1 contradicts the fact that |D| ≤ 29, as required. Hanani (1960) proved that a necessary and sufficient condition for the existence of a 3-(v, 4, 1) design is that v ≡ 2 or 4 (mod 6). Lemma 3.2.4. If there exists a 3-(v, 4, 1) design, then v ≡ 2 or 4 (mod 6). v−2 ) 2. (v3). blocks. The (43) ×4 = vr, shows that v is even. Moreover, as a 2-design, we have integrality of v−2 4 2 (v−1)(v−2) . Since gcd(v − 1, v − 2) = 1, if 2|(v − 1) and 3|(v − 2), then and then r = 6 v = 2m + 1 and v = 3l + 2 for some m and l, and hence, v = 6k + 5; similarly if 2|(v − 2), v = 6k + 4; similarly if 6|(v − 1) (or 6|(v − 2)), then v = 6k + 1 (or v = 6k + 2 respectively). However, v is odd contradicting the fact v is even. It follows that v ≡ 2 or 4(mod 6).. Proof. Each 3-(v, 4, 1) design is also a 2-(v, 4,. design with. 4. Similar arguments 3-(v, 4, 1) work well for designs with even number of vertices. 8.

(15) Theorem 3.2.5. If Y is a set of k points of a 3-(2k, 4, 1) design containing no block, then the complement of Y contains no block either. , blocks, and n1 = (2k−1)(2k−2) Proof. For a 3-(2k, 4, 1) design, it has b = 2k(2k−1)(2k−2) 6 24 n2 = k − 1, n3 = 1, where ni is the number of blocks containing a fixed set of i points. Suppose there exists a set Y of k points containing no blocks, but Y containing a block, say B1 , without loss of generality, we may assume that Y = {1, 2, . . . , k}, and the transpose of the first column of its incidence matrix is {0, 0, 0, 0, . . . , 0, 1, 1, 1, 1, 0, . . . , 0}, then Y ∩ B1 = ∅. Let D = {Bi : Y ∩ Bi 6= ∅, i = 1, 2, . . . , b}, then |D| ≤ b − 1 because Y ∩ B1 = ∅. On the other hand, the principal of inclusion and ¡k ¢ ¡k ¢ = b, × 1 = k(k−1)(2k−1) × (k − 1) + − exclusion shows that |D| = k × (2k−1)(2k−2) 6 3 2 6 contradicts the fact that |D| ≤ b − 1, as required. This following result, known as the Friendship Theorem, is due to Erdös, Rényi and Sós(1966), as referenced by van Lint [9, pp.45]. Theorem 3.2.6. Let Γ be a graph in which any two vertices have a unique common neighbor, then either Γ is a windmill, or Γ is regular (and hence strongly regular). A similar result works for a family of subtrees of a tree. Theorem 3.2.7. Let T1 , . . . , Tk be subtrees of a tree T such that the trees Ti and Tj have a vertex in common for all 1 ≤ i < j ≤ k. Then T has a vertex that is in all the Ti . 1st proof. We first claim that Ti ∩ Tj is a subtree of T for any i 6= j. Suppose, to the contradictory, that there exist p, q ∈ Ti ∩ Tj such that p and q are not connected in Ti ∩ Tj . There exist ti ∈ Ti and tj ∈ Tj such that p, ti , q, tj , p are on a cycle, a contradiction. Without loss of generality, suppose |T1 ∩ T2 | is the maximal intersection number For trees T1 and T2 , we show that Ti ∩ (T1 ∩ T2 ) is nonempty for each i. Let T1 ∩ T2 be a tree T 0 . Suppose to the contradictory, Ti ∩ T 0 is empty. Let x ∈ (T1 ∩ Ti − T2 ), and y ∈ (T2 ∩ Ti ) − T1 . Thus, there exists t0 ∈ T 0 such that x, t0 and y are on a cycle, a contradiction. We further show that (Ti ∩ T 0 ) ∩ (Tj ∩ T 0 ) is nonempty for distinct i and j. Suppose to the contradictory that it is empty. Let x ∈ (Ti ∩ Tj ) − T 0 , then 1. x, ti , t0i , t1 , t0j are on a cycle where ti ∈ Ti , t0i ∈ Ti ∩ T 0 , and t1 ∈ Ti whenever x∈ / T1 ∪ T2 , or. 9.

(16) 2. x, ti , t0i , t1 , t0j are on a cycle where ti ∈ Ti , t0i ∈ Ti ∩ T 0 , t1 ∈ T1 (resp. T2 ) and t0j ∈ Tj ∩ T 0 whenever x ∈ T1 (resp. T2 ). Therefore Ti ∩ T 0 is a subtree of T 0 , i = 3, 4, . . . , k, and (Ti ∩ T 0 ) ∩ (Tj ∩ T 0 ) 6= ∅, the condition is the same as original assumption, by this way, we finally obtain that T1 ∩ T2 ∩, . . . , ∩Tk is a subtree of T . 2nd proof. If there exists a subtree Tk with |V (Tk )| = 1, then it is trivial. Assume there is no subtree with only one vertex. We prove it by introduction on |V (T )| = n. Since the case n = 2 is trivial, we suppose that T has a vertex that in all the Ti , i.e., ∩ki=1 Ti 6= ∅, for n ≤ m. When n = m + 1, we choose a leaf x of T . If x ∈ ∩ki=1 Ti , then ∩ki=1 Ti 6= ∅. If x ∈ / ∩ki=1 Ti . then deleting x from the tree T , and we obtain a tree called T 0 such that |V (T 0 )| = m. Let Ti0 = Ti − {x}. T10 , T20 , . . . , Tk0 are subtrees of T 0 so that Ti0 and Tj0 have a vertex in common for all 1 ≤ i < j ≤ k. then ∩ki=1 Ti0 6= ∅. So ∩ki=1 Ti 6= ∅. By induction, T has a vertex in all the Ti . 3.3. Combinatorial Information derived from Quadratic functions. Quadratic functions sometimes can be associated with some combinatorial problems, and their discriminants usually provide some inequalities with combinatorial interests. This technique will be illustrated in section in terms of a few typical examples. Theorem 3.3.1 ([9],pp.). Each block of a 2-(v, k, λ)design meets nontrivially at least k(r − 1)2 (k − 1)(λ − 1) + (r − 1) other blocks. The sufficient and necessary conditions for the equal sign holds is that (k − 1)(λ − 1) + (r − 1) points, and hence it is any two blocks intersect in precisely r−1 a symmetric design. Proof. Let B be a block of B. Let a = |{Bi : Bi 6= B and |Bi ∩ B| 6= 0}|, and k X ni be the number of blocks meet B in precisely i points. Therefore ni = a, i=1 k X. ini = kr − 1, and hence. i=1. k X. i(i − 1)ni = k(k − 1)(λ − 2) and. i=1. X (i − x)2 ni = ax2 − 2k(r − 1)x + k((k − 1)(λ − 1) + (r − 1)) ≥ 0. So its discriminant D = (−2k(r − 1))2 − 4ak((k − 1)(λ − 1) + (r − 1)) 10.

(17) 2. k(r−1) as required. Further more, the equal sign is negative, and hence a > (k−1)(λ−1)+(r−1) holds if and only if the quadratic equation k X (i − x)2 ni = ax2 − 2k(r − 1)x + k((k − 1)(λ − 1) + (r − 1)) = 0 i=1. has a unique solution x = x = i.. 2k(r−1) 2a. =. (k−1)(λ−1)+(r−1) , r−1. i.e., ni > 0 if and only if. An n-arc of a symmetric 2-(v, k, λ)design Π = (X, B) is a set of n points such that no three of which are contained in a block. A block B of Π is called a tangent to an n-arc S if |S ∩ B| = 1. An n-arc A is called an oval of type I if each point of A lies on a unique tangent, and an oval of type II if it has no tangents. Theorem 3.3.2 ([9],pp.). Let A be a type I oval in a symmetric 2-(v, k, λ) design with k − λ even. Then any point of the design lies on either one or all tangents to A. Proof. Observe that k, λ and n are all odd, and so any point lies on at least one tangent to A. We apply a different version of the ’variance trick’. Let ni be the P P P number of points which lie on i tangents. Then ni = v, ini = nk, and i(i − P 1)ni = n(n − 1)λ. Therefore (i − 1)(i − n)ni = 0, whence every point lies on one or all the tangents. Theorem 3.3.3 ([15],pp.232). Let Γ be a strongly regular graph with parameters (v, k, λ, µ), connected complement and Spec(Γ) = (k 1 , θmθ , τ mτ ). If k < mθ , then (mθ − 1)(kλ − λ2 − (k − mθ )τ 2 ) − (λ + (k − mθ )τ )2 > 0. Proof. Consider the quadratic polynomial p(x) p(x) = (mθ − 1)(ka − λ2 − k − mθ x2 ) − (λ + (k − mθ )x2 ) = (mθ − 1)kλ − mθ λ2 + 2λ(mθ − k)x + (k − 1)(mθ − k)x2 , we find that its discriminant is −4a(mθ − k)(mθ − 1)k(k − 1 − λ). Since k < m and 1 < m, we see that this is negative unless λ = 0. If λ = 0, then p(x) = (k − 1)(mθ − k)x2 , and consequently p(τ ) 6= 0, unless τ = 0. If λ = 0 and τ = 0, then Γ is the complete bipartite graph Kk,k with eigenvalues k, 0, and −k. However, if τ = 0, then θ = −k and m = 1, which contradicts the condition that k < mθ .. 11.

(18) Orthogonal arrays provide a convenient method of obtaining a sequence of pseudorandom sample points. Suppose that A is an orthogonal array OA(k, n) on the set X with |X| = n. A Two-point sampling is accomplished as follows: let r be a random row in A, use the k values A(r, 1),. . . , A(r, k) along the rth row of A as the k sample points, note that these k samples points are not necessarily all distinct. If the rows of A are indexed by X × X, then a random row of A is specified by choosing two points independently at random from X. An elementary combinatorial analysis of the two-point sampling technique which allows us to calculate a bound on the error probability is presented below. Suppose that I is a yes-instance, and define the set S of witnesses (note that we do not know the set S explicitly). S = {x ∈ U : f (I, x) = 1}, We have |S| = m = (1 − ²)n. km + m(k − 1) n Let ai denote the number of rows of A in which there are exactly I occurrences of elements from S. Call a row of the matrix a bad row if none of the elements in the row is a witness. Then the error probability is simply the probability that the randomly selected row is a bad row. Hence, the error probability, when we run the algorithm A using k ample points chosen from a random row of A, is a0 (1) err(S) = 2 n As mentioned above, we do not know the set S explicitly. However, an upper bound on the error probability can be obtained by computing err = max{err(A) : ¡ ¢ }. Since an OA(k, n) has n2 rows, A∈ X m. Theorem 3.3.4 ([35],pp.96-99). err(S) ≤ 1 −. n X. a i = n2 .. (2). i=0. Counting the number of occurrences of witnesses in A in two ways, there are exactly ai rows in which there are i occurrences of witnesses, in any column of A. It follows that each point occurs exactly n times. n X. iai = knm.. (3). i=0. Similarly, counting the number of occurrences of pairs of witnesses occurring in the same row in two ways, it yields: n X. i(i − 1)ai = k(k − 1)m2 .. i=0. 12. (4).

(19) For a real number x, we have 0≤. n n n n n X X X X X (i − x)2 = (i2 − 2xi + z 2 )ai = i2 ai − 2x iai + x2 ai i=1. i=1. i=1. = k(k − 1)m2 + knm − 2xknm + x2. n X. i=1. i=1. ai .. i=1. It follows from equations (2), (3) and (4), that n X. ai ≥. i=1. 2knmx − knm − k(k − 1)m2 x2. The right hand side of (5) is maximized when x = n X i=1. ai ≥. n+(k−1)m . n. (5) Hence, we get. kmn2 . n + (k − 1)m. (6). Now, from (2) and (6), we have a 0 ≤ n2 −. knm2 . n + (k − 1)m. Then, we get the following bound on the error probability from (1): err(S) ≤ 1 −. km . n + (k − 1)m. Note that this bound on the error probability approaches 0 only linearly quickly as a function of k. A small example which meets the bound is given by the following OA(3, 4): 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 2 3 0 1 3 2 1 0 0 1 2 3 1 0 3 2. 13.

(20) 4. Quasi-symmetric designs and Strongly Regular designs. We are interested in 2-(v, k, λ) designs with constraints over intersections between their blocks. Those of symmetric designs, quasi-symmetric designs, semi-symmetric design or even quasi-semi-symmetric designs. Based on the paper [29], the class of quasi-symmetric designs with specific blocks will be surveyed in Section 4-1. Based on the papers [14] by T. S. Fu and Y. Huang, the class of quasi-semi-symmetric designs will be surveyed in Section 4-2. 4.1. Quasi-symmetric designs with Good Blocks. The family of quasi-symmetric designs form a broad class of 2-designs containing all affine and symmetric designs and linear spaces. Quasi-symmetric designs not in these subclasses are sparse. One motive for investigating quasi-symmetric designs with prescribed geometric conditions, such as the existence of subdesigns, is that perhaps a new construction might become apparent. The structure of quasisymmetric designs which have special blocks on which are induced quasi-symmetric designs will be surveyed in this section, we show that-with the exception of linear, affine, and projective spaceswthere are only two possible parameter sets for such designs. Only one example is known of a design of one parameter set type and none of the other. In this section, quasi-symmetric designs with (x, y) = (0, y) or (1, y) are studied. Definition 4. Let x and y be non-negative integers with x ≤ y. 2-design is quasisymmetric with intersection numbers x and y if |B ∩ C| ∈ {x, y} for any two distinct blocks B and C and both intersection numbers are realized. The design is proper if x 6= y; otherwise it is improper. It is well-known that a 2-(v, k, λ) design with λ > 1 is symmetric if and only if any two distinct blocks meet in λ points. A design is resolvable if its blocks can be partitioned into subsets (parallel classes) each of which partitions the whole point set. The partition of blocks is called a parallelism. Blocks in the same parallel class are parallel. Two distinct parallel blocks are disjoint. If, further the number of points common to any two nonparallel blocks is constant, the design is said to be affine. Clearly, affine designs are examples of quasi-symmetric designs with 0 as an intersection number. Linear spaces, i.e., 2-(v, k, 1) designs, are also examples. Theorem 4.1.1 ([29]). For any quasi-symmetric design with intersection numbers x and y: 1. y − x divides both k − x and r − λ if the design is proper; 14.

(21) 2. (b − 1)xy + k(k − 1)(λ − 1) = k(r − 1)(x + y − 1); and 3. (k − 1)(λ − 1) = (r − 1)(y − 1) whenever x = 0. Similar to affine planes and to symmetric designs, the notion of good blocks for quasi-symmetric designs is introduced. Lemma 4.1.2 ([29]). Let Π = (X, B) be a quasi-symmetric 2-(v, k, λ) design with intersection number x and y, where x < y. Let B and C be blocks meeting in y points. 1. The intersection of any two distinct blocks containing B ∩ C is B ∩ C. 2. Any point not in B ∩ C is on at most one block containing B ∩ C. v−y blocks containing B ∩ C. Equality holds if and only if 3. There are at most k−y any point not in B ∩ C is on a unique block containing B ∩ C.. Proof. 1 follows since y ≥ x and |B ∩ C| = y. Since two distinct blocks cannot have y + 1 or more common points, then 2. follows. Finally, 3. is an easy deduction from 1. and 2. Definition 5. Let Π be a quasi-symmetric 2-(v, k, λ) design, a block B is good if there is a (necessarily unique) block containing both p and B ∩ C whenever a block C meeting B in y points and any point p not in B ∩ C. Note that if λ = 1 or, equivalently, y = 1 then all blocks in Π are good. The v−k will be proved to be a useful parameter for a quasi-symmetric quotient m = k−y 2-(v, k, λ) design. The next lemma is a simple consequence of the definition of a good block and lemma 4.1.2. Lemma 4.1.3 ([29]). Let Π be as in Lemma 4.1.2 and let B be any block. Then B is a good block if and only if for any block C meeting B in y points there are exactly m + 1 blocks containing B ∩ C. v−k must be a positive integer for any quasiAs an easy consequence, m = k−y symmetric design with a good block. Those quasi-symmetric designs having 0 as an intersection number and having a good block are classified parametrically as shown in the following theorem. We show that a quasi-symmetric design with intersection number 1 and y > 1 and a good block belongs to one of three types: Examples of quasi-symmetric designs which are not symmetric designs or affine designs or linear spaces are rather rare, so construction methods for quasi-symmetric designs are of interest. The classification. 15.

(22) problem for quasi-symmetric designs, even in case x = 0 appears to be a difficult open problem. One approach in the study of such designs is to impose some additional parametric or structural condition. Another approach is to consider substructures and induced substructures of such quasi-symmetric designs with a possible view to obtain new construction methods. They consider quasi-symmetric designs with x = 0 and having a certain type of block, referred to as a good block. According to them, in any quasi-symmetric design with intersection numbers x, y (0 ≤ x ≤ y), a block B is said to be good, if for any block C with |B ∩ C| = y and any point p ∈ / B ∩ C, there is a (unique) block containing p and B ∩ C. It is clear that if λ = 1 (or, equivalently, y = 1) in a quasi-symmetric design, then all blocks are good. Their notion of good blocks in quasi-symmetric designs extends the earlier notion for affine designs and symmetric designs. One of the main results of [29] is the following theorem. Theorem 4.1.4 ([29]). Let Π = (X, B) be a quasi-symmetric 2-(v, k, λ) design with intersection numbers x = 0 and y. Suppose that Π has a good block. Then Π is a symmetric design or an affine design or a linear space or else has one of the following two parameter sets: 1. v = y 4 (y 3 − y 2 + 2y − 1), k = y 2 (y 2 − y + 1), λ = y 3 + y + 1; 2. v = y(y 4 − y 3 + y 2 − y + 1), k = y(y 2 − y + 1), λ = y 2 + 1. Those QSD in case 1. and 2. Π induces a 2-(k, y, 1) design on the points of any good block B, the blocks of the induced design being distinct non-empty intersections of B with the other blocks of Π. It is remarked in [29] that the smallest possible value of y in case 1. is y = 3, the value y = 2 being excluded by the non-existence of a projective plane of order 10. It is also noted that the unique 2-(22, 6, 5) Witt design having x = 0, y = 2 satisfies case 2., but no examples with y ≥ 3 are known. Theorem 4.1.5 ([29]). Let Π be a proper quasi-symmetric 2-(v, k, λ) design with λ > 1 and having an intersection number 0. Let B be a good block and let Π0 be v−k . If there is ) design induced on B, where m = k−y the quasi-symmetric 2-(k, y, λ−1 m a good block which is disjoint from B, then Π0 is resolvable. Theorem 4.1.6 ([29]). Let Π be a quasi-symmetric 2-(v, k, λ) design with λ > 1 and having an intersection number 0. Then the following statements 1. and 2. are equivalent: 1. All blocks of Π are good. 16.

(23) 2. Either Π is the design of points and hyperplanes of projective or affine space, or all lines (see definition below) of Π have size y and Π has parameter of type 1. or 2. in Theorem 4.1.4. Let Πp denote the incidence structure whose points and blocks are the lines and blocks of Π on a point p, respectively. In case 2.(ii), Πp is a projective plane of order y 3 + y or y 2 for any point p. If y = 2, Π is the unique Witt 3-(22,6,1)design. Proper quasi-symmetric designs with x = 1 and which have a good block is reviewed in this section. The following are two well-known examples of such designs. Example: 1. Let P G(4, q) be the four-dimensional projective geometry over GF (q). Let Π = P G2 (4, q), respectively. Then Π is a quasi-symmetric 2-(v, k, λ) design 3 −1 5 −1 , x = 1 and y = q + 1. All blocks of Π are good. , k = λ = qq−1 with v = qq−1 2. The unique 2-(23, 7, 21) Witt design Π, which is also a 4-(23, 7, 1) design, is a quasi-symmetric design with x = 1 and y = 3. All blocks in this design are good since y = 3 and there is exactly one block on any four points. Theorem 4.1.7 ([28]). Let Π = (X, B) be a proper quasi-symmetric 2-(v, k, λ) design with intersection numbers 1 and y and with a good block. Then Π is one of the following: 1. a quasi-symmetric design with parameters 3 −1 5 −1 , b = (q 2 + 1)v, r = (q 2 + 1)k, x = 1 and y = q + 1 , k = λ = qq−1 v = qq−1 (q ≥ 2) as those of P G(4, q). 2. The unique 2-(23, 7, 21) Witt design with v = 23, k = 7, λ = 21, b = 253, r = 77, x = 1, y = 3. 3. A quasi-symmetric design with parameters v = 1 + ((α − 1)λ + 1)(y − 1) and k = 1 + α(y − 1), for some integer α > y ≥ 5, and in which the design Π1 induced on a good block is a 2-(k, y, 1) design. No example of designs satisfying case 3. of the theorem has been found. Theorem 4.1.8 ([28]). Let Π = (X, B) be a quasi-symmetric design, with parameters as those of P G2 (4, q) and with (x, y) = (1, q + 1). All blocks of Π are good if and only if Π is isomorphic to P G2 (4, q).. 17.

(24) 4.2 4.2.1. A Class of Strongly Regular Designs SRD Quasi-semi symmetric designs. Motivated by the study of the geometric structures associated with the half dual polar graph Dn,n (q) and the alternating forms graphs Alt(n, q), some specific conditions over incidence structures were considered by Fu and Huang [14]: (QSS1) every two distinct points are in 0 or λ common blocks, (QSS2) every two distinct blocks intersect in 0 or µ points, (QSS3) if λ = 1, then there are constants k and r such that every block contains k points and every point is on r blocks, (QSS4) if (x, B) is a nonincident pair of point x and block B, then there are exactly α blocks of x intersecting B. Let λ, µ, and α be positive integers. A finite incidence structure Π = (X, B) is called a quasi-semi-symmetric design (abbreviated ’QSSD’) for λ, µ with nexus if conditions (QSS1)-(QSS4) are satisfied. Clearly, λ = 1 if and only if µ = 1, and hence Π is a semilinear space or a partial linear space (see Brouwer et al., 1989, for the definition). Condition (QSS3) is necessary to ensure the k-uniformity and r-regularity of Π (i.e., every blocks of Π contains k points, and every point of Π is in r blocks). An example that satisfies (QSS1) and (QSS2) with λ = µ = 1 but does not satisfy (QSS3) is given in Huang an Pan (1988). Partial geometries, first studied by Bose, are examples of QSSDs with λ = µ = 1, and partial λ-geometries, introduced by Cameron and Drake (1980) are QSSDs with λ = µ. QSSDs with multiple intersections, i.e., λ ≥ µ ≥ 2, were treated []. Basic properties, associated combinatorial structures, some examples constructed from vector spaces, and some existence conditions for QSSDs with µ = λ − 1 ≥ 2 are described. Two extremal conditions that provide an upper bound and a lower bound, respectively, for α. The following two equivalent conditions, called the (*)-conditions, were studied for (s, r; µ)-nets in Huang and Laurent (1993) and for partial λ-geometries 2 under these extremal conditions. in Cameron and Drake (1980). Indeed, α = λ (µ−1) µ ⊥ For a nonflag (x, B), |x ∩ B| is a constant β, where βλ = αµ, and we let Πx,B be the incidence structure defined over x⊥ ∩ B. The structure of Πx,B , together with the (*)-condition, gives a sharp lower bound for β (and hence for α). Symmetric designs, semisymmetric designs, and partial λ-geometries are among such structures. In this paper, in addition to some general properties, we study the existence conditions for QSSDs with µ = λ − 1 ≥ 2 and the properties of 18.

(25) QSSDs satisfying the following extremal condition: if B1 and B2 are two blocks with a nonempty intersection, then there are another λ − 2 blocks B3 , . . . , Bλ such 2 under such a condition, that ∩1≤i≤λ Bi = B1 ∩ B2 . We show that α ≥ λ (µ−1)+λ µ and QSSDs with equality are classified whenever µ = λ or µ = λ − 1 following a classification of affine polar spaces by Cohen and Shult (Geometraic Dedicata 35 (1990), 43-76). 4.2.2. Strongly Regular Designs. The notion of strongly regular designs was first introduced by D.G. Higman as a class of 1-design arising in the investigation of coherent configurations of small types. Indeed, SRD’s are 1 12 -designs in the sense of Neumaier [31] and form a self-dual class. An SRD has a point graph and a block graph of which are strongly regular. An incidence structure Π = (X, B) with a point-block incidence matrix M is called a strongly regular design if there exists nonnegative integers ai , bi , Ni , Pi and Si , i = 1, 2, such that 1. JM = S1 J, M J = S2 J, 2. M M T = S2 I + a2 A1 + b2 (J − I − A1 ), M T M = S1 I + a1 A2 + b1 (J − I − A2 ), 3. A1 M = N1 M + P1 (J − M ), M A2 = N2 M + P2 (J − M ), where A1 and A2 are square matrices of orders v and b, respectively. Note that A1 and A1 = J − I − A1 , respectively, A2 and A2 = J − I − A2 , form the adjacency matrices of a pair of complementary strongly regular graphs. These conditions can be interpreted in terms of the relationship between points and blocks as follows: 1. each block consists of k points, and point lies in r blocks; 2. any two points lie in either a2 or b2 common blocks, and any two blocks meet in either a1 or b1 points; 3. for a pair of a point x and a block B, x is collinear with either P1 points or N1 points of B1 depending on whether x is in B or not, the number of blocks containing x and meeting B is either P2 or N2 depending on whether x ∈ B or not. We will focus on those strongly regular designs with b1 = b2 = 0, i.e., we are concerned with those strongly regular designs with incidence matrix M satisfying the following conditions: JM = kJ, M J = rJ, M M T = rI +aA1 , M T M = kI +bA2 , A1 M = aM + b(J − M ) and M A2 = N2 M + P2 (J − M ) (note that k for S1 , r for 19.

(26) S2 , A for A1 , B for A2 ). This class of strongly regular designs is called a quasisemisymmetric design with nexus, and it was used as a geometric framework for a characterization of alternating bilinear forms graphs of order 4. An SRD(v, k, a, b) is called resolvable if the blocks can be partitioned into classes such that each class form a partition of X and any two blocks have common points if they are in different classes. Lemma 4.2.1. For a resolvable SRD(v, k, a, b), then 1. for a pair (x, B) of nonincident point x and block B, there is a unique block points on B, on x and parallel to B, and hence x is collinear with b(r−1) a points when2. two distinct points x and y are collinear with k − 2 + (r−a)(br−b−a) a2 br(r−1) points; ever they are collinear, or otherwise they are collinear with a2 and hence its point graph is a strongly regular graph on v points with k = r(k − 1)/a, f =, andg = r(r − 1)b/a2. 4.2.3. Some properties of SRD(v, k, λ, µ). Lemma 4.2.2 ([14]). Let Π = (X, B) be a SRD(v, k, λ, µ), µ ≥ 2, with nexus α. The following two conditions are equivalent: 6 ∅, then there exist B3 , . . . , Bλ ∈ 1. if B1 , B2 are two distinct blocks, with B1 ∩B2 = B such that ∩1≤i≤λ Bi = B1 ∩ B2 , which consists of µ points. 2. if B1 , B2 , B3 are three distinct blocks with |B1 ∩ B2 ∩ B3 | ≥ 2, then |B1 ∩ B2 ∩ B3 | = µ. Corollary 4.2.3 ([14]). Let (x, B) be a nonflag of a SRD satisfying the (*)condition, and let A1 and A2 be two distinct blocks of x intersecting B. Then |A1 ∩ A2 ∩ B| ≤ 1. Lemma 4.2.4 ([14]). Let Π = (X, B) be a SRD(v, k, λ, µ), g ≥ 2, satisfying the (*)-condition with nexus α and let (x, B) be a nonflag. Then 1. β ≥ λ(µ − 1) + 1, and hence α ≥. λ2 (µ−1)+λ , µ. 2. equality holds if and only if the structure Πx,B is a 2-(λ(µ − 1) + 1, µ, 1) design. Substituting µ = λ (= q +1) and µ = λ−1 (= q) in the previous lemma, we have α ≥ q 2 + q + 1 and α ≥ q 2 + q, respectively. Examples (iii) and (iv) in the previous section show that both bounds are sharp. Moreover, the 2-designs mentioned above in the QSSDs of Examples (iii) and (iv) are projective planes and affine planes of order q, respectively. 20.

(27) An upper bound for β (and hence for α) is obtained by the following extremal condition, called the (∆)-condition. Any three distinct pairwise collinear points are in at least one common block. Lemma 4.2.5 ([14]). Let Π = (X, B) be a SRD(v, k, λ, µ), g ≥ 2, satisfying the 2 . (∆)-condition with nexus α. Then β ≤ λ(µ − 1) + 1, and hence α ≤ λ (µ−1)+λ µ Corollary 4.2.6 ([14]). Let Π = (X, B) be a SRD(v, k, λ, µ), satisfying the (*)2 . Then the (∆)-condition holds. condition with nexus α = λ (µ−1) µ For a nonflag (x, B), the incidence structure Πx,B is determined under the (*)and (∆)-condition. Corollary 4.2.7 ([14]). Let Π = (X, B) be a SRD(v, k, λ, µ), µ ≥ 2 with nexus α. The following are equivalent: 1. Π satisfies the (*)- and (∆)-conditions, 2. Πx,B is a 2-(λ(µ − 1) + 1, µ, 1) design. Cameron and Drake (1980) showed that a SRD(v, k, λ, µ) satisfying the (*)condition with nexus α = λ2 − λ + 1 is obtained from a polar space of type D4 (q) with one family of maximal totally isotropic subspaces as the block set. As a result, its point graph is isomorphic to D4,4 (q). In this section, we shall prove a similar result for a SRD(v, k, λ, λ − 1) with nexus α = λ2 − λ. 2 Let Π = (X, B) be a SRD(v, k, λ, µ) satisfying the with nexus α = λ (µ−1)+λ µ (i.e., β = λ(µ − 1) + 1). Associate Π with an incidence structure Π0 = (X, L) with a collection P of substructures, where L = {A ∩ B : A, B ∈ B are distinct with A ∩ B 6= ∅} and let P = {x⊥ ∩ B : x ∈ X, B ∈ B, x ∈ / B}. Members of L and P are called lines and planes, respectively. Clearly, the point graphs of Π and Π0 are identical. For any two collinear points x and y, let A1 , . . . , Aλ be the blocks containing x and y and denoted by xy the line A1 ∩ A2 = ∩1≤i≤λ Ai (by the (*)-condition). Since α reaches the lower bound, the (∆)-condition also holds, by Corollary 4.2.6. Thus x⊥ ∩ y ⊥ = ∪1≤i≤λ Ai , and {Ai − xy : 1 ≤ i ≤ λ} forms a partition of x⊥ ∩ y ⊥ − xy. Hence the incidence structure Π0 = (X, L) is a gamma space, and each block of Π induces a maximal singular subspace in Π0 (refer to Brouwer et al., 1989, for the definitions of gamma spaces and singular subspaces). Note also that each plane in P is a singular subspace too. A triple of points is called a triangle if they are pairwise collinear but not contained in a common line. The main theorem in this section is as follows:. 21.

(28) Theorem 4.2.8 ([14]). Let Π = (X, B) be a SRD(v, k, λ, µ) satisfying the (*)2 (i.e., β = λ(µ − 1) + 1). Then condition with α = λ (µ−1)+λ µ 1. if µ = λ (= q + 1 ≥ 3), then Π0 = (X, L) is the polar space of type D4 (q) and the point graph of Π is isomorphic to D4,4 (q). 2. if µ = λ − 1 (= q ≥ 4), then either Π0 = (X, L) is the affine polar space of type D4 (q) − ∞⊥ and the point graph of Π is isomorphic to Alt(4, q), or k = 55 , 115 . Construction for families of resolvable SRD with parameters (v, k, a, b) = (q 3 , q 2 , 2, q), (q 3 , q 2 , 2q , q) for q = 2n a power of 2. As a consequence, a few families of other designs as well as strongly regular graphs are derived. Among others, this gives a strongly regular graph SRG(196, 135, 94, 90) which is new to the list of such graphs up to 280 vertices complied by A.E. Brouwer in the CRC Handbook of Combinatorial Designs.. 22.

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