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A Study of Strongly Regular Multigraphs with Some

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Student: Yuning Chen

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A Thesis

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

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for the Degree of

Master

in

Applied Mathematics

June 2005

Hsinchu, Taiwan, Republic of China

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A Study of Strongly Regular Multigraphs with

Some Applications

Student: Yuning Chen

Advisor: Tayuan Huang

Department of Applied Mathematics

National Chiao Tung University

June 2005

Abstract

The conception of strongly regular multigraph was first proposed by Bose in 1976, followed by Neumaier and Metsch in 1982 and 1995 respectively for the problem of embedding of quasi-residual 2-design. In particular, Neumaier asserted that the collinearity graph of a unique 11

2-design if it meets some

constraints over its parameters.

The spectral properties of strongly regular multigraphs are studied in Sec-tion 3, we show that they can be characterized as multigraphs with exactly three distinct eigenvalues, we show further when they are strongly regular graphs in terms of their eigenvalues. For reference purpose, the results to-gether with the arguments for the proofs of the papers of Bose, Neumaier and Metsch are summarized is Section 4. A class of strongly regular multigraphs associated with the alternating forms graphs is studied in Section 5. Un-der some numerical constraints, they are the collinearity graphs of uniquely determined 11

2-designs, which provide some information for the geometric

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Contents

2d¿b iii

Abstract iv

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1 Introduction 1

2 Definitions and Preliminaries 5

3 Spectral of Strongly Regular Multigraphs 11

4 A Survey of Papers of Bose, Neumaier and Metsch 17

4.1 The 1976 paper of Bose . . . 18

4.2 The 1982 paper of Neumaier . . . 19

4.3 The section 2 of 1995 paper of Metsch . . . 25

4.4 The section 3 of 1995 paper of Metsch . . . 30

5 A Class of Strongly Regular Multigraphs 33

1

Introduction

The notion of strongly regular multigraphs (SRMG) was first introduced by R. C. Bose [2] but in a very cumbersome notation. While characterizing quasi-residual

2-designs, A. Neumaier [9] gave an equivalent definition of strongly regular multigraphs

in an elegant and self-contained way with some improvements over some results; the concept ”type” by five parameters (m, n, µ, γ, R).

Recently, Metsch [7] continued the study of embeddings of residual 2-designs within the framework of strongly regular multigraphs. However, no specific example were given in the papers mentioned above.

For a 2-(v, k, λ)-design π = (X, B), it is known that every point is in r = λ(v−1)_k−1 blocks, and the number of blocks in B is b = λv(v−1)_k(k−1) and the number of points is

v = k +n(k−1)_λ where n = r − λ is called the order of the design. Moreover, Fisher’s in-equality b > v holds, with in-equality if and only if every pair of distinct blocks intersects in λ points. A 2-(v, k, λ)-design with b = v, i.e., v = 1 + k(k−1)_λ is called a symmetric

design. If B is a block of a symmetric 2-(v, k, λ)-design π = (X, B), two associated

designs, i.e., the derived design Bder and the residual design Bres with respect to the block B arise naturally. In both cases, respect to the block B arise naturally. In both cases, the blocks are the members of B − {B}, and the points are the points in B for

Bder, and the points not in B for Bres, incidences are remained the same as before. In terms of the order n, the derived design is a 2-(n + λ, λ, λ − 1)-design, and the residual design is a 2-(w, n, λ)-design with w = n(n+λ−1)_λ .

The block multigraph of a 2-design π = (X, B) is the multigraph defined over the block set B, and two distinct vertices A, B are connected by mA,B = |A ∩ B| edges.

Neumaier showed that the block multigraph of a 2-(v, k, λ)-design of order n is a strongly regular multigraph SR(m, n, µ, γ, R) with

SR(m, n, µ, γ, R) = (k, n, k2_{λ, k(k − 1)(λ − 1), k(n + λ − 1)).}

A partial converse is given in the following theorem with some constraints over its parameters:

Theorem ([9], Theorem 1.1) Every strongly regular multigraph with parameters

SR(m, n, µ, γ, R) = (k, n, k2_{λ, k(k − 1)(λ − 1), k(n + λ − 1))}

for positive integers n, k 6= 1, λ, and

n > max{k(k − 1)λ2_{− (k − 1)}2_λ,

2(k − 1)(k2_{λ + kλ − 2λ + 1),}1

2(k2− 1)(k2λ − k + 2)}

is isomorphic to the block multigraph of a 2-(v, k, λ)-design with v = k + (r−λ)(k−1)_λ .

Its proof involves more general designs, namely 11

2-designs (called partial geometric designs in [2]), and weak 11

2-designs (without assuming constant block size). Note that

2-designs, dual 2-designs, transversal designs, semiregular partially balanced incom-plete block designs, partial geometries, and polar spaces are examples of 11

2-designs,

see Neumaier [9]. The notion of partial geometric design D(r, k, t, c) was introduced as a generalization of a partial geometry (r, k, t) (with c = 0 above). A partial geomet-ric design D(r, k, t, c) gives rise in a natural manner to a strongly regular multigraph (SRMG) G(D) whose parameters depend on r, k, t and c, as a generalization of

strongly regular graphs (SRG). The block multigraphs of 11

2-designs, and dually, the

point multigraphs of weak 11

2-designs still are strongly regular, and by investigating

charac-terization theorems which specialize to Theorem 1.1([9]).

The matrix techniques were used by Neumaier in order to get the relations among the five parameters, and then to derive the essential relations between strongly regular multigraph and 11

2-design.

For any two distinct blocks A, B 6= H, denote by αA,B the number of points in H incident with A and B, and by βA,B the number of points not in H incident with A and B. Then αA,B+ βA,B = λ, in particular βA,B 6 λ. Moreover, the multigraph on

B − {H}, with αA,B edges between A and B, is the block multigraph of Bder. Hence, the residual design satisfies the conditions given in the following theorem.

Theorem ([9], Theorem 1.2) A quasi-residual 2-(w, n, λ)-design π is embeddable if and only if the following conditions are satisfied:

(1) Any two distinct blocks A and B intersect in αA,B6 λ points.

(2) Let G be the multigraph defined on the blocks with αA,B = λ−βA,B edges between

A and B, then

a. G is a strongly regular multigraph SR(m, n, µ, γ, R) with

SR(m, n, µ, γ, R) = (λ, n, λ2_{(λ − 1), λ(λ − 1)(λ − 2), λ(n + λ − 2));}

b. G is isomorphic to the block multigraph of a 2-(n + λ, λ, λ − 1)-design π0_.

Theorem ([9], Theorem 1.3) A quasi-residual 2-(w, n, λ)-design is embeddable if either

(1) λ = 3, and n > 76, or (2) λ 6= 3, and n > 1

2(λ2− 1)(λ3− λ2− λ + 2).

a polynomial function f (λ) of degree 5 such that every quasi-residual 2-(v, k, λ) design is residual provided that k > f (λ). It was further improved that

f (λ) = (λ2_{− 1)(λ}3_{− λ}2 _{− λ + 2) by Neumaier [9] in Theorem 1.3, and then} f (λ) = (_√8

3λ + λ + 5)λ

2_{(λ − 1) by Metsch [7].}

This embedding theorem will be a consequence of more general characterization theorems for certain strongly regular multigraphs (see Theorem 2 [9] and its corollary in the introduction).

As to us, Neumaier’s most important contribution is the Theorem 4.4 written in his paper. He showed that if Γ is a strongly regular multigraph SR(m, n, µ, γ, R) with integral m > 2, integral µ ≡ 0 mod m, µ > 0, and

n > max{m − 1 + (µ + m)γ m2 , 2(m − 1)(µ + 1 − m) + 2γ, m(m − 1) 2 (µ + 1) + m γ 2 + m − 1} then Γ is the point multigraph of a unique 11

2-design, with parameters

(r0_{, k}0_{, t}0_{, c}0_{) = (m,}R m + 1, µ m, γ m).

As we know, strongly regular graph has some necessary and sufficient conditions. Two necessary and sufficient conditions on strongly regular multigraphs were given in Section 3. These papers of R. C. Bose, Neumaier, Metsch over three decades will be surveyed in Section 4, together with the technique and arguments used by them. We pay more attention for the unique theorem (Theorem 4.4 [9]) for its unique presentation of 11

2-designs. A class of specific example associated with alternating bilinear form

2

Definitions and Preliminaries

In this section, we define strongly regular multigraphs and 11

2-designs. Then we give

the necessary and sufficient conditions of strongly regular multigraphs and 11

2-designs.

Finally, we will give a theorem which will be useful in Section 5.

Definition 2.1 A simple graph Γ with the vertex set V and with the edge set

E is called a strongly regular graph (SRG) with parameters (v, k, λ, µ), denoted by SRG(v, k, λ, µ), if (1) |V | = v, and (2) for x, y ∈ V |N(x) ∩ N(y)| =    k if x = y λ if x ∼ y µ if x  y .

A multigraph Γ contains a nonempty set V of vertices and a set E of edges. For all x, y ∈ V = V (Γ), mx,y := number of edges joining x and y, and define mx,x := 0.

Definition 2.2 A multigraph Γ is called a strongly regular multigraph with parame-ters (m, n, µ, γ, R), if:

(1) P_y∈V mx,y = R for each x ∈ V ;

(2) P_x∈V ma,xmb,x = (n − 2m)ma,b + m(n − m)δa,b + µ, where δa,b = 1 if a = b, otherwise, δa,b = 0.

(3) P_y∈V mx,y(mx,y− 1) = γ for each x ∈ V .

Here m, n, µ, γ and R are real numbers with n > 0.

Note thatP_y∈V(mx,y)2 = P

y∈V mx,y+ P

y∈V mx,y(mx,y− 1) and from (1)∼(3), we have

If γ =P_y∈V mx,y(mx,y− 1) = 0 in definition 2.2, then either mx,y = 0 or mx,y = 1 for x, y ∈ V , and hence Γ is a simple graph, and moreover Γ is a strongly regular graph with parameters

(v, k, λ, µ) = (R(R − n + 2m − µ − 1)

µ + R − 1, R, n − 2m + µ, µ).

Note that R =P_y∈V mx,y, and γ = P

y∈V mx,y(mx,y− 1) for x, y ∈ V were defined explicitly in the definition, but there is no explicit definition on m, n and µ. Such a definition is very unnatural. Combinatorial interpretations of m, n, µ are interest-ing for us. The parameter µ in SRMG(m, n, µ, γ, R) is identical with that of µ in

SRG(v, k, λ, µ) in case γ = 0.

The notion of strongly regular graphs can be stated in terms of the matrix.

Lemma 2.3 Let A be the adjacency matrix of a simple graph Γ, then the follow-ing are equivalent:

(1) Γ is a strongly regular graph.

(2) AJ = kJ, A2 _{= (λ − µ)A + (k − µ)I + µJ.}

Similar to strongly regular graphs, we want to know the matrix expression of strongly regular multigraph.

Lemma 2.4 Let A be the adjacency matrix of a multigraph Γ of order v, then the following are equivalent:

(2) AJ = RJ, A2 _{= (n − 2m)A + m(n − m)I + µJ, with real numbers R, m, n, µ, and} n > 0, and v = (R+m)(R+m−n)_µ .

Some subsets of vertices including claws, maximal cliques play essential roles in the study of their structures. We will explain in Section 4 that m is the constant number of maximal cliques containing a fix vertex in the study of maximal claws.

Definition 2.5

(1) A clique is a set of mutually adjacent vertices. A maximal clique is a clique not properly contained in any other clique.

(2) A claw (x, A) consists of a vertex x and an anticlique A such that x is adjacent to every vertex of A. The order of the claw (x, A) is defined by P_y∈Amx,y.

It is well knows that the block graph of a quasi-symmetric 2-design is strongly regular. This leads to the question that whether some strongly regular multigraphs associated with some designs of various types? A class of incidence structure lies be-tween 1-designs (regular) and 2-designs is defined. We will show in Section 4 that the collinearity graphs of this class of incidence structures are indeed strongly regular multigraphs.

For an incidence structures, let mx,y = number of blocks containing points x and

y, and define mx,x = 0.

Definition 2.6 A 11

2-design with parameter (r, k, t, c) is an incidence structure I =

(P, B) such that

(1) each point x lies on r blocks; (2) each block l contains k points;

(3) for a point x and a block l

a. t =P_y∈lmx,y > 1 is a constant if x ∈/ l; b. c =P_y∈l−{x}(mx,y− 1) is a constant if x ∈ l.

Let A be the incidence matrix of the incidence structure under consideration, and (x, B) is a pair of point and block, note that

AAT_{A(x, B) =}X_{A(x, B}0_{)A(y, B}0_{)A(y, B)}

is the number of the pair (y, B0_{) such that x ∈ B}0 _{and y ∈ B ∩ B}0_{. The following} matrix expression for 11

2-designs is immediate:

Lemma 2.7 For a binary matrix A, the following are equivalent: (1) A is the incidence matrix of a 11

2-design.

(2) AJ = rJ, JA = kJ and AAT_{A = (r + k − 1 + c − t)A + tJ for some integers r, k,}

t, c with t > 1.

Lemma 2.8 Each 2-(v, k, λ) design is a 11

2-design with parameters

(r, k, t, c) = (λ(v − 1)

k − 1 , k, kλ, (k − 1)(λ − 1)).

Conversely, each 11

2-design with parameters (r, k, t, c) satisfying (t + 1 − c − k)k = t is

a 2-(v, k, λ) design where

(v, λ) = (1 + r(k − 1)

λ , t + 1 − c − k).

Proof:

(1) Since the incidence matrix A of a 2-(v, k, λ) design satisfies

AJ = rJ, JA = kJ, and AAT _{= (r − λ)I + λJ,}

AAT_{A = (r − λ)A + λkJ; and c is computed by AA}T_{A = (r + k − 1 + c − t)A + tJ.} (2) Let A be the incidence matrix of a 11

2-design with (t + 1 − c − k)k = t, and let X = AAT _{− (r + k + c − t − 1)I + (k + c − t − 1)J.}

Then we have X2 _{= 0, and hence X = 0 because X is a symmetric matrix. Hence} AAT _{= (r + k + c − t − 1)I − (k + c − t − 1)J = (r − λ)I + λJ,}

with λ = t + 1 − c − k. Q.E.D.

Lemma 2.9 ([9], Theorem 3.2) The collinearity graph of a 11

2-design with

param-eters (r, k, t, c) is a strongly regular multigraph with paramparam-eters (m, n, µ, γ, R) with (m, n, µ, γ, R) = (r, k + r + c − 1 − t, rt, rc, r(k − 1)).

The above lemma, shows that the collinearity graph of a 11

2-design is a strongly

regular multigraph. Following this trend, we are interested in those strongly regu-lar multigraphs which are the collinearity graph of 11

2-designs or of even unique 1 1 2

-designs? The following Theorem of Neumaier provides sufficient numerical constrains to guarantee the uniqueness of such 11

2-designs. Its proof will be given in Section 4.

Theorem 2.10 ([9], Theorem 4.4) If Γ is a strongly regular multigraph with pa-rameters (m, n, µ, γ, R) with m > 2, integral µ ≡ 0 mod m, µ > 0, and

n > max{m − 1 + (µ + m)γ m2 , 2(m − 1)(µ + 1 − m) + 2γ, m(m − 1) 2 (µ + 1) + m γ 2 + m − 1} then Γ is the collinearity multigraph of a unique 11

2-design, with parameters

(r, k, t, c) = (m, R m + 1, µ m, γ m).

There is no example of strongly regular multigraphs meeting those numerical con-straints found in the papers of Bose, Neumaier and Metsch.

In the final section, we use the symmetric association scheme to define a distance regular graph, and defined a class of strongly regular multigraphs by giving the mul-tiedge on the induce subgraph of this distance regular graph.

Definition 2.11 An association scheme with d classes is a finite set X together with d + 1 relations Ri on X such that

(1) {R0, R1, ..., Rd} is a partition of X × X; (2) R0 = {(x, x)|x ∈ X};

(3) for each i ∈ {0, 1, ..., d} there exists a j ∈ {0, 1, ..., d} such that (x, y) ∈ Ri implies (y, x) ∈ Rj;

(4) for any (x, y) ∈ Rk the number pkij of z ∈ X with (x, z) ∈ Ri and (z, y) ∈ Rj depends only on i, j and k;

(5) pk

ij = pkji for all i, j, k ∈ {0, 1, ..., d}. Moreover, if (3) and (5) replaces by

(3’) if (x, y) ∈ Ri, then also (y, x) ∈ Ri, for all x, y ∈ X and i ∈ {0, 1, ..., d}. Then it is called the symmetric association scheme.

Definition 2.12 A distance regular graph is a simple graph with the intersection numbers {b0, b1, ..., bd−1; c1, c2, ..., cd} satisfying the follows:

For all (x, y) ∈ V , if ∂(x, y) = i then (1) ci := |Γ1(x) ∩ Γi−1(y)|,

(3) ai := |Γ1(x) ∩ Γi(y)|, and

If ∂(x, y) = 0 then ki := |Γi(x) ∩ Γi(y)|.

3

Spectral of Strongly Regular Multigraphs

The matrix expressions in terms of their adjacency matrix of strong regular graph and strongly regular multigraph are quite similar. We are interested to know the spectrum of strongly regular multigraphs?

The eigenvalues of strongly regular graphs can be easily calculated in terms of the matrix equation of its adjacency matrix. We also know that the strongly regular graph are those connected regular graph with exactly three distinct eigenvalues, and the spectral of Γ is as follows:

spec(Γ) = (k1_{, (}1 2((λ − µ) + √ ∆))m1_{, (}1 2((λ − µ) − √ ∆))m2_), where ∆ = (λ − µ)2_{+ 4(k − µ), and} m1 = (v − 1) + 2k − (v − 1)(λ − µ) √ ∆ m2 = (v − 1) +2k + (v − 1)(λ − µ)√ ∆

The following Lemma will prove that the converse is also true.

Lemma 3.1 A connected k -regular graph Γ is a strongly regular graph with pa-rameters (v, k, λ, µ) if and only if it has exactly three distinct eigenvalues k > θ1 > θ2.

Moreover, (θ1, θ2) = (1₂((λ−µ)+ √ ∆), 1 2((λ−µ)− √ ∆)) where ∆ = (λ−µ)2_{+4(k −µ).}

Lemma 3.1 can be extended to strongly regular multigraphs with a mimic proof. Before shown the Lemma 3.3, we need some technique. First, we need to make sure if Γ is a connected R-regular multigraph, then R is also an eigenvalue. And next we need make sure the multiplicity of the corresponding eigenvalue R is 1.

Proposition 3.2 Let Γ be a multigraph,

(1) Γ is R-regular multigraph if and only if the largest absolute eigenvalue of Γ is R. (2) The multiplicity of R as an eigenvalue is 1 if Γ is connected.

Proof: Let A be the adjacency matrix of Γ. Take x = (x1, x2, ..., xv)T be an eigen-vector for eigenvalue λ, and let xi be a coordinate of largest absolute value among coordinates of x. For the i-th coordinate of Ax, we have

|λ||xi| = |(Ax)i| = | v X j=1 Aijxj| 6 | v X j=1 Aijxi| 6 | X i∼j Aij||xi| = R|xi|.

Hence λ 6 R. Equality requires xj = xi for all xj ∈ N(xi). We can iterate this argument to reach all coordinates for vertices in Γ. Thus, the multiplicity of R is

1. Q.E.D.

Lemma 3.3 A connected R-regular multigraph Γ is a strongly regular multigraph with parameters (m, n, µ, γ, R) if and only if it has exactly three distinct eigenvalues

R > θ1 > θ2. Moreover, spec(Γ) = (R1, (n − m)t1, (−m)t2) where

(t1, t2) = ( m(v − 1) − R n , (n − m)(v − 1) − R n ). Proof:

First, we assume that Γ is a strongly regular multigraph with an adjacency matrix

A3 _{= (n − 2m + R)A}2_{− (R(n − 2m) − m(n − m))A − mR(n − m)I, that is,} A3_{− (n − 2m + R)A}2_{+ (R(n − 2m) − m(n − m))A + mR(n − m)I = 0.}

Hence the minimal polynomial of A is given by

f (x) = (x − R)(x − (n − m))(x + m).

So the strongly regular graph has three distinct eigenvalues R, n − m, and −m, let 1,

t1, and t2 be their multiplicities respectively. Since the trace of A equal to the sum of

all eigenvalues, and the number of eigenvalues are equal to the number of vertices. So we have (t1, t2) = ( m(v − 1) − R n , (n − m)(v − 1) − R n ).

On the other hand, let R > θ1 > θ2 be the three distinct eigenvalues, since the

mul-tiplicity of R is 1 because A is the adjacency matrix of a connected regular multigraph. Define

M := 1

(R − θ1)(R − θ1)

(A − θ1I)(A − θ2I).

Since A and A2 _{so symmetric. Take x is an eigenvector of A with the corresponding}

eigenvalues θ1 (or θ2 respectively), then Mx = 0, i.e., x is an eigenvector of M with

eigenvalues 0. Thus all eigenvectors of corresponding eigenvalue is 0 are in the kernel of M. Thus, the rank of M is 1, equal to the multiplicity of R. Then we have

M =      a1 a2 · · · av a1 a2 · · · av ... ... ... ... a1 a2 · · · av      or      a1 a1 · · · a1 a2 a2 · · · a1 ... ... ... ... av av · · · av     

Furthermore, M is symmetric, hence a1 = a2 = · · · = av = a is a constant. Let ~1 be the all one vector, then we have M~1 = ~1. Hence M = 1

vJ, that is A2 is a linear combination of A, J and I. Thus A is the adjacency matrix of some strongly regular

Remark: From above proof, we have n − m < R because the multiplicity of R is 1.

Form Lemma 3.1 and Lemma 3.3, we know that a connected k -regular graph (respectively, multigraph) with exactly three distinct eigenvalues k, θ1 and θ2 is a

strongly regular graph (respectively, a strongly regular multigraph). Moreover, 1 vJ = 1 (k − θ1)(k − θ2) (A − θ1I)(A − θ2I), that is, A2 = (θ1 + θ2)A − θ1θ2I + 1 v(k − θ1)(k − θ2)J.

Consider that (θ1+θ2)A−θ1θ2I +1_v(k −θ1)(k −θ2)J = (n−2m)A+m(n−m)I +µJ

for some m, n, µ, γ with n > 0 are real numbers then we can compute it directly and then have (m, n, µ, γ, R) = (−θ2, θ1− θ2, (k − θ1)(k − θ2) v , (k − θ1)(k − θ2) v − k − θ1θ2, k).

If the graph is the simple graph, then we have (v, k, λ, µ) = ((k − θ1)(k − θ2)

k + θ1θ2

, k, θ1θ2+ θ1+ θ2+ k, k + θ1θ2).

Since γ = P_y∼xmx,y(mx,y− 1) = (k−θ1)(k−θ_v 2)− k − θ1θ2 > 0, we have (k − θ1)(k − θ2) > v(k+θ1θ2). The case equality or otherwise correspond to strongly regular graphs

or strongly regular multigraphs respectively.

Note: As γ = 0, Γ is a strongly regular graph with parameters (v, k, λ, µ) = (R(R − n + 2m − µ − 1)

Then we have (m, n, µ, γ, R) = (1 2( √ ∆ − (λ − µ)),√∆, µ, 0, k), where ∆ = (λ − µ)2_{+ 4(k − µ).} Then spec(Γ) = (R1, (n − m)t1_{, (−m)}t2_), where (t1, t2) = ( m(v − 1) − R n , (n − m)(v − 1) − R n ),

can be reduce to the strongly regular graph with

spec(Γ) = (k1, (1 2((λ − µ) + √ ∆))m1_{, (}1 2((λ − µ) − √ ∆))m2₎ where ∆ = (λ − µ)2_{+ 4(k − µ) and} (m1, m2) = ((v − 1) + 2k − (v − 1)(λ − µ) √ ∆ , (v − 1) + 2k + (v − 1)(λ − µ) √ ∆ ).

Then we have the following:

Theorem 3.4 If A is an adjacency matrix of order v of a k -regular (either simple or multiple) graph with three distinct eigenvalues k > θ1 > θ2, then (k − θ1)(k − θ2) > v(k + θ1θ2). Moreover,

(1) if (k − θ1)(k − θ2) = v(k + θ1θ2), then the graph is a strongly regular graph SRG(v, k, λ, µ) with

(v, k, λ, µ) = ((k − θ1)(k − θ2)

k + θ1θ2

, k, θ1θ2+ θ1+ θ2+ k, k + θ1θ2);

(2) if (k − θ1)(k − θ2) > v(k + θ1θ2), then the graph is nontrivial strongly regular

multigraph SRMG(m, n, µ, γ, R) with (m, n, µ, γ, R) = (−θ2, θ1− θ2, (k − θ1)(k − θ2) v , (k − θ1)(k − θ2) v − k − θ1θ2, k).

Proof: From the definition γ =P_y∼xmx,y(mx,y− 1), we have (k − θ1)(k − θ2) > v(k + θ1θ2).

(1) If γ = 0, then A must be the adjacency matrix of a strongly regular graph. (2) If γ > 0, then A must be the adjacency matrix of a strongly regular multigraph. Q.E.D.

A comparison between strongly regular graphs and strongly regular multigraphs are included in the following table:

SRG(v, k, λ, µ) SRM G(m, n, µ, γ, R)

adjacency matrix a symmetric (0,1)-matrix a symmetric matrix_{with nonnegative entries} adjacency AJ = kJ, AJ = RJ,

matrix expression A2_{= (λ − µ)A + (k − µ)I + µJ A}2_{= (n − 2m)A + m(n − m)I + µJ}

relative design quasi-symmetric 2-design 11 2-design three distinct k, 1 2((λ − µ) ± √ ∆), where R, n − m, −m eigenvalues ∆ = (λ − µ)2_{+ 4(k − µ)} v =(k−θ1)(k−θ2) k+θ1θ2 , m = −θ2, n = θ1− θ2 given k = k, µ =(k−θ1)(k−θ2) v k > θ1> θ2 λ = θ1θ2+ θ1+ θ2+ k, γ = (k−θ1)(k−θ_v 2)− k − θ1θ2, µ = k + θ1θ2 R = k (k − θ1)(k − θ2) = v(k + θ1θ2) (k − θ1)(k − θ2) > v(k + θ1θ2) Table 1

4

A Survey of Papers of Bose, Neumaier and Metsch

The subjects of strongly regular multigraph were thoroughly studied in the papers of R. C. Bose ([2], 1976), Neumaier ([9], 1982) and Metsch ([7], 1995). A study of these three papers will be given in this section in an unified way.

These three papers share some common ground with various terminologies, there facts are listed in Table 2 for reference. The results, and the ways of their proofs for each papers were given in subsections 4-2∼4-4 respectively.

The content of subsections 4-2∼4-4 are in the following order: First, the main theorems concerning the quasi-residual designs are gives; following by their proofs in sketch the strategies for the proofs of these main theorem and provided in figures 1∼3.

The relation of Bose, Neumaier, and Metsch’s theorem:

Bose Neumaier Metsch

[2], Lemma 2.1 (lower bound) [9], Lemma 2.3 (upper bound) [2], Theorem 2.8 (unique)

[2], Theorem 2.5 [9], Lemma 4.1(b)

[2], Theorem 2.6 [9], Lemma 4.1(c)

[2], Theorem 2.7 [9], Lemma 4.1(c)

[2], Theorem 3.3(1) [7], Lemma 3.22

[2], Theorem 3.3(2) [9], Lemma 4.2 [7], Lemma 2.10

[2], Theorem 3.3(3) [7], Lemma 3.10 [2], Lemma 3.3 [7], Lemma 3.11 [2], Theorem 4.2 [9], Theorem 3.5 [2], Theorem 4.1 [9], Theorem 1.2(2) [2], Theorem 4.3 [9], Theorem 1.3 Table 2

4.1

The 1976 paper of Bose

A multigraph G is said to be regular of degree n and loop degree d if n(x) := P

y∼xmx,y = n and d(x) := P

y6=x

(mx,y)(mx,y−1)

2 = d are constant for each vertex x

in G.

A regular multigraph G is called edge regular if for any set of adjacent vertices

x and y, the quantity p(x, y) := P_z∼x,ymx,zmy,z depends only on the multiplicity

mx,y of x and y. The concept of ”type” was defined by Bose over edge regularity of multigraphs.

An edge regular multigraph will be said to be of the type Gk{r, d; α0, α1, . . . αr} if it satisfies the following properties:

(1) mx,y 6 r for any edge xy.

(2) The degree n(x) of any vertex is given by n(x) = r(k − 1). (3) The loop degree d(x) = d , for any vertex x.

(4) For any edge xy for which mx,y = m > 1, p(x, y) = m(k − 2) is a divisor of αm. (5) If the vertices, x and y are nondajacent, i.e., mx,y = 0, p(x, y) 6 α0.

Furthermore, he proposed the definition of strongly regular multigraphs over the edge regular graphs. Obviously, the difference between edge regular and strongly reg-ular multigraph is that the edge regreg-ular posed condition over pairs of adjacent vertices depends on the multiplicity mx,y, but strongly regular multigraphs posed conditions over pair of distinct vertices depends on the multiplicity mx,y.

Then Bose gave a definition of a claw. A claw (x, S) of the multigraphs Gk is defined as a set of vertices S = {y1, y2, . . . , ys} nonadjacent to each other and each

adjacent to a vertex x, i.e., mx,yi = mi > 1 and myi,yj = 0 if i 6= j = 1, 2, . . . , s. A

claw (x, S) is said to be of type (a1, a2, . . . , ar) if ai be the number of edges xy, y ∈ S for which mx,y = i. With the claw we can associate four parameters (s, µ, δ, π) define as follows:

(1) s =Pr_i=1ai, the order of the claw. (2) µ =Pr_i=1iai =

P_s

i=1mi, the multiplicity of the claw. (3) δ = 1

2

P_r

i=1i(i − 1)ai = 12

P_s

i=1mi(mi− 1), the loop multiplicity of the claw. (4) π =Pr_i=1aiαi =

P_s

i=1αmi, the coefficient of edge regularity of the claw.

He defined grand cliques and then studied the structures strongly regular multi-graph in term of their claws and grand cliques.

4.2

The 1982 paper of Neumaier

For any two distinct blocks A, B 6= H, where H is a block, denote by αA,B the number of points in H incident with A and B, and βA,B the number of points not in H incident with A and B.

Theorem 4.2.1 ([9], Theorem 1.2) A quasi-residual 2-(v, k, λ)-design π = (P, B) is embeddable if and only if the following three conditions are satisfied:

(1) Any two distinct blocks A and B intersect in βA,B 6 λ points.

(2) The multigraph G defined over the blocks, with αA,B = λ − βA,B edges between A and B, is a strongly regular multigraph with parameters

(m, n, µ, γ, R) = (λ, k, λ2_{(λ − 1), λ(λ − 1)(λ − 2), λ(k + λ − 2)).}

Theorem 4.2.2 ([9], Theorem 1.3) A quasi-residual 2-(v, k, λ)-design π = (P, B) is embeddable if either λ = 3, and k > 76, or λ 6= 3, and k > 1

2(λ2−1)(λ3−λ2−λ+2).

The main theorem shows that under what numerical constraints, a strongly regular multigraph will be the point graph of a unique 11

2-design (Theorem 4.2.17). To prove

it, Neumaier first showed that this strongly regular multigraph is the point graph of a

weak 11

2-design, and then showed further that it is a 112-design. The uniqueness of this

11

2-design is guaranteed by showing that each block is a grand clique of the strongly

regular multigraph under consideration.

Step 1: Show first that there is no s-claw (x, S) whenever s > m in terms of the quantity N =P_x6=y(αx− my,x)(αx− my,x − 1) where αx :=

P

y∈Smx,y, and the con-tradictory argument related to upper and lower bounds (Lemma 4.2.14).

Step 2: Show further that each point is in exactly m grand cliques, and each edge ab of multiplicity ma,bis in exactly ma,bcliques by a constructive argument (Lemma 4.2.15).

Step 3: Show that a SRMG is a point graph of a weak 11

2-design if and only if

the two condition satisfied.

(1) there is a collection P of cliques such that every point is in exactly m cliques of P

, and

(2) every edge ab of multiplicity ma,b is in exactly ma,b cliques of P

(Theorem 4.2.12) in terms of incidence matrices of designs and the adjacency matrices of strongly regular multigraphs.

Step 4: Show the constant size of blocks (Theorem 4.2.13) under either of the fol-lowing conditions

(1) two distinct points are in at most one blocks,

(2) t = λ(λv+n)_r is an integer with λ(n + 1 − r) < (1 − λ)(t + 1) in terms of the quantity

s(x, B) = |{(a, A)|(a, A) ∈ P × B, x, a ∈ A, a ∈ B}|.

Lemma 2.8 Theorem 4.2.9 Corollary 4.2.10 Lemma 4.2.4 Lemma 4.2.3 Theorem 4.2.11 Lemma 4.2.6 Lemma 4.2.15 Lemma 4.2.14 Lemma 4.2.16 Theorem 4.2.12 Theorem 4.2.13 Theorem 4.2.17 Figure 1: [9], Neumaier

Lemma 4.2.3 ([9], Lemma 2.1) Let A be an integral symmetric matrix with zero diagonal satisfying AJ = RJ, A2 _{= (n − 2m)A + m(n − m)I + µJ. If n > max{2m −}

4, 2m − 1 + µ + γ} where γ = m(n − m) + µ − R, then A is the adjacency matrix of a strongly regular multigraph with parameters (m, n, µ, γ, R).

Lemma 4.2.4 ([9], Lemma 2.2) Let Γ be a strongly regular multigraph graph with parameters (m, n, µ, γ, R).

(1) m − n 6 ma,b 6 m.

(2) m > 1, with equality if and only if Γ is the disjoint union of complete graphs. (3) If there are nonadjacent points then n > m.

(4) µ > (R + m)(m − n), with equality if and only if ma,b = m − n for all a 6= b. (5) µγ 6 (n − 2m + µ)(m(n − m) + µ), with equality if Γ contains no triangles. (6) µ > 2m − n.

(7) If n 6 2m + 4, then γ < 2m(n − m) + n − 2m − 1 + µ.

Definition 4.2.5 A maximal clique C with |C| > n

2+µ+1−m is called a grand clique.

Lemma 4.2.6 ([9], Lemma 2.3) An edge of multiplicity one is in at most one grand clique.

Definition 4.2.7 An incidence structure with an incidence matrix A is a weak 2-design if AJ = rJ, AAT _{= nI +λJ and a weak 1}1

2-design if AJ = rJ and AATA = nA+λJA.

Theorem 4.2.8 ([9], Theorem 3.2) The point multigraph of a weak 11

2-design with

parameters (v, n, r, λ) is a strongly regular multigraph with parameters

(m, n, µ, γ, R) = (r, n, λ(λv + n), λ(λ − 1)v − r(r − 1) + (r + λ − 1)n, λv − n + r). In particular, the point multigraph of a 11

2-design with parameters (r, k, t, c) is a

strongly regular multigraph with parameters

(m, n, µ, γ, R) = (r, r + k + c − 1 − t, rt, rc, r(k − 1)) and (r, k, t, c) = (m, R m + 1, µ m, γ m).

Theorem 4.2.9 ([9], Theorem 3.3) The block multigraph of a 2-(v, k, λ)-design of order n is a strongly regular multigraph with parameters

(m, n, µ, γ, R) = (k, n, k2_{λ, k(k − 1)(λ − 1), k(n + λ − 1)).}

Corollary 4.2.10 ([9], Corollary 3.4) Two distinct blocks A and B of a 2-(v, k, λ)-design intersect in at least k − r + λ points.

Theorem 4.2.11 ([9], Theorem 3.5) Let π = (P, B) be a quasi-residual 2-(v, k, λ)-design with k > 2λ3_{− 4λ}2_{+ 4λ − 1. Then}

(1) two distinct blocks intersect in at most λ points, and

(2) the multigraph Γ on the blocks, with αA,B = λ − βA,B edges between A and B, is a strongly regular multigraph with parameters

(m, n, µ, γ, R) = (λ, k, λ2_{(λ − 1), λ(λ − 1)(λ − 2), λ(k + λ − 2)).}

Theorem 4.2.12 ([9], Theorem 3.6) A strongly regular multigraph Γ with parameters (m, n, µ, γ, R) is the point multigraph of a weak 11

2-design if and only if there is a

collection P of cliques such that every point is in exactly m cliques of P, and every edge ab of multiplicity ma,b is in exactly ma,b cliques of

P

. Moreover, the blocks are the cliques ofP, and the weak 11

2-design has parameters

(v, n, r, λ) = ((R + m)(R + m − n)

µ , n, m,

µ R + m).

Theorem 4.2.13 ([9], Theorem 3.7) Let π = (P, B) be a weak 11

2-design with

param-eters (v, n, r, λ), and λ < 1. If either

(1) two distinct points are in at most one blocks, or

(2) t = λ(λv+n)_r is an integer with λ(n + 1 − r) < (1 − λ)(t + 1), then π = (P, B) is a 11

2-design, with parameters

(r, k, t, c) = (r,λv + n

r ,

λ(λv + n)

r , n + 1 + t − r − k).

Lemma 4.2.14 ([9], Lemma 4.1) Let Γ be a strongly regular multigraph with param-eters (m, n, µ, γ, R) with µ > 1, and integral m > 2. The following hold:

(1) If 2n > m(m − 1)(µ + 1) + mγ + 2m − 2, then s 6 m for every s-claw.

(2) If 2n > (m − 3)(µ − m) + 2γ + 2m − 2 then every s-claw (1 6 s 6 m − 2) can be extended to a (s + 1)-claw.

(3) If there are no s-claw with s > m, then every (m − 1)-claw is in at least n − 1 − (m − 2)(µ + 1 − m) − γ many of m-claws.

(4) If (a, S) is a maximal m-claw, then there are at least m(n − 2) − (m − 2)µ − 2γ many of m-claws (a, S0_{) such that |S ∩ S}0_{| = m − 1.}

Lemma 4.2.15 ([9], Lemma 4.2) Let Γ be a strongly regular multigraph with pa-rameters (m, n, µ, γ, R) with µ > 1, and integral m > 2. If there are no s-claws with

s > m and if

n > max{1

2(m − 3)(µ − m) + γ + m − 1, 2(m − 1)(µ + 1 − m) + 2γ}

then each edge ab of multiplicity ma,b is exactly ma,b cliques, and each point is in exactly m grand cliques.

Lemma 4.2.16 ([9], Lemma 4.3) If Γ is a strongly regular multigraph with parameters (m, n, µ, γ, R) with µ > 1, and integral m > 2, and

n > max{2(m − 1)(µ + 1 − m) + 2γ,1

2m(m − 1)(µ + 1) + 1

2mγ + m − 1} then Γ is the point multigraph of weak 11

2-design with parameters (v, n, r, λ) given by

(v, n, r, λ) = ((R + m)(R + m − n)

µ , n, m,

µ R + m).

Theorem 4.2.17 ([9], Theorem 4.4) If Γ is a strongly regular multigraph with pa-rameters (m, n, µ, γ, R) with integral m > 2, integral µ ≡ 0 mod m, µ > 0, and

n > max{m − 1+(µ + m)γ m2 , 2(m − 1)(µ+1− m)+2γ, m(m − 1) 2 (µ +1)+ m γ 2+ m− 1}

then Γ is the point multigraph of unique 11

2-design, with parameters

(r, k, t, c) = (m, R m + 1, µ m, γ m).

Corollary 4.2.18 ([9], Corollary 4.5) Every strongly regular multigraph with param-eters

(m, n, µ, γ, R) = (k, n, k2λ, k(k − 1)(λ − 1), k(n + λ − 1))

such that n, k, λ are positive integers, k 6= 1 and

n > max{k(k − 1)λ2_{− (k − 1)}2_{λ, 2(k − 1)(k}2_{λ + kλ − 2λ + 1),}1

2(k

2_{− 1)(k}2_{λ − k + 2)}}

is isomorphic to the block multigraph of a 2-(v, k, λ)-design with v = k + λ−1_{n(k − 1).}

Theorem 4.2.19 ([9], Theorem 4.6) If Γ is a strongly regular multigraph with pa-rameters (m, n, µ, γ, R) with µ > 1, integral m, 2 6 m 6 n, and

n > max{2(m − 1)(µ + 1 − m),m(m − 1)

2 (µ + 1) + m − 1} then Γ is the point graph of a unique partial geometry with parameters

(r, k, t) = (m,R

m + 1, µ m).

4.3

The section 2 of 1995 paper of Metsch

Theorem 4.3.1 ([7], Theorem 2.1) A strongly regular multigraph with parameters (m, n, µ, γ, R) with µ, m > 2. Suppose

(2) 2(a + 1 − m)n > a(a − 1)µ + aγ + 4m(a + 1) − 2m2_{− (a + 1)(a + 2)}

for some a > m, then the family M = {C|C is a maximal clique with |C| > n − 2m − (a − 2)µ − 3

2γ + a + 1} satisfies

(a) For each vertex x and if ax is maximum order of a claw (x, A), then m 6 ax 6 a and there exists exactly ax cliques in M which contains x.

(b) If x, y are adjacent, then x and y lies in at least mx,y and at most 2mx,y− 1 cliques of M.

(c) If x, y are adjacent and x lies in exactly m cliques of M, then there exists exactly

mx,y cliques in M contains x and y.

Definition 4.3.2 A maximal clique C with |C| > n − 2m − (a − 2)µ − 3

2γ + a + 1 is

called a normal clique.

To derive each pair of adjacent vertices x and y are contained in exactly mx,y nor-mal cliques (Lemma 4.3.9) whenever the maxinor-mal of order ax of a claw (x, A) is m. It will then be used to ensure condition (c) for Theorem 4.4.1.

Toward this goal, we first study lower bounds and upper bounds for the number of normal cliques containing a vertex (indeed exactly bound, Lemma 4.3.1 (1)), or containing a pair of adjacent vertices respectively (Lemma 4.3.4) in terms of some functions for counting purpose.

a. each vertex is adjacent to at least R − 1

2γ vertices;

b. each vertex x is adjacent to at most 1

2γ vertices y with mx,y > 2;

c. any two distinct vertices x, y have at least wx,y − 3₂γ common neighborhoods, where wx,y =

P

For a maximal claw (x, A) with order ax, and a point y ∈ A adjacent to x:

We first show that there are at most 2mx,y− 1 cliques containing adjacent vertices

x and y (Lemma 4.3.8 (2)) by applying the principle of inclusion and exclusion to

derive upper bound and lower bound of P2mx,y

i=1 |Ci − {x, y}| respectively in terms of the quantity P2mx,y

i=1 |Ci| for 2mx,y maximal cliques containing x and y, together with a hypothesis condition in Theorem 4.3.1.

If mx,y = 1, Metsch showed that

Cy := {x, y} ∪ {z|z ∈/ A ∈ A, x ∼ y, y ∼ z, ∀w ∈ A − {y}, w  z},

lies in exactly one normal clique (Lemma 4.3.7 (1)) by showing the size of Cy meeting the require condition for normal cliques in terms of the maximality of (x, A) and the bounds (Lemma 4.3.4) for neighbors. We then show that Cy meets A nontrivially (Lemma 4.3.6). Indeed, it is true for any normal clique containing x by considering upper bounds of |C|.

On the other hand, for y ∈ A with mx,y > 1, then A can be replaced by another maximal claw (x, A0_{) with A}0 _{= A−{y}∪{z}

1, z2, . . . zmx,y} with order ax and mx,zi = 1

for each i (Lemma 4.3.7 (2)) by showing the existence of exactly pairwise nonadjacent

mx,y common neighbors of x and y outside A and not adjacent to any vertices in A except y in terms of the bound for neighbors (Lemma 4.3.4). Continuing this pro-cess, any maximal claw (x, A) can be replaced by another maximal claw (x, A0_{) with}

mx,z = 1 for each z ∈ A0.

There is a normal clique Cy meeting A0 nontrivially for each y ∈ A0, it follows that there are ax normal cliques containing x. We conclude that there exist exactly

ax cliques contains x for all x ∈ V (Lemma 4.3.8 (1)) and then there are at least mx,y normal cliques containing adjacent pair x, y of points (Lemma 4.3.8 (1)).

Finally, we claim that ax > m (Lemma 4.3.5 (2)) by considering |{z|z ∼ x, z ∼ y ∈

A}| + |A|. We conclude that each pair of adjacent vertices x and y are contained in

exactly mx,y normal cliques (Lemma 4.3.9) by combining Lemma 4.3.8 (1) and Lemma 4.3.5 (2). The hypotheses of Theorem 4.3.1 is guaranteed under the condition given in Lemma 4.3.3.

Lemma 4.3.3 ([7], Lemma 2.11) If m > 2, µ > m, γ + 1 and n > 2(1 + _√2

3)mµ = 2(3+2√3)

3 mµ ≈ 4.3mµ, then the hypotheses of Theorem 4.3.1 are satisfied for the unique

integer a satisfying _√2

3m − 1 < a 6 2

√

3m.

Theorem 4.3.5 is proved by Lemmas 4.3.4 ∼ 4.3.9, and the relationship between Lemma 4.3.4 ∼ Lemma 4.3.9 are given below:

Lemma 4.3.5 (2)

Lemma 4.3.4 Lemma 4.3.7 (1) Lemma 4.3.9 Lemma 4.3.5 (1) Lemma 4.3.6 Lemma 4.3.8 (1)

Lemma 4.3.8 (2) Lemma 4.3.7 (2)

Figure 2: Section 2 in [7], Metsch

Lemma 4.3.4 ([7], Lemma 2.2)

(1) Each vertex x is adjacent to at most 1

2γ vertices y with mx,y > 2;

(2) Each vertex is adjacent to at least R − 1

(3) Any two distinct vertices x, y have at least wx,y− 3₂γ common neighborhoods.

Lemma 4.3.5

(1) Every claw has order at most a. ([7], Lemma 2.4)

(2) Every maximal claw has order at least m. ([7], Lemma 2.9)

Lemma 4.3.6 ([7], Lemma 2.6) If (x, A) is a maximal claw, and C is a normal clique containing x, then A ∩ C 6= φ.

Lemma 4.3.7 For a claw (x, A) of order ax and a vertex y ∈ A: (1) if mx,y = 1, then the set,

C := {x, y} ∪ {z ∈/ A adjacent y but not to any vertex of A − {y}}

is contained in a normal clique. ([7], Lemma 2.5)

(2) if s := mx,y > 2, then there are mutually non-adjacent vertices y1, y2, . . . ys satis-fying mx,yi = 1 and such that each yi is adjacent to y but not to any other vertex

of A. ([7], Lemma 2.7)

Lemma 4.3.8 ([7], Lemma 2.8)

(1) Each vertex x lies on exactly ax normal cliques.

(2) Any two adjacent vertex x and y lies in at most 2mx,y − 1 cliques ([7], Lemma 2.3), and lies in at least mx,y normal cliques.

Lemma 4.3.9 ([7], Lemma 2.10) For a claw (x, A) with maximum order m, then

4.4

The section 3 of 1995 paper of Metsch

Theorem 4.4.1 ([7], Theorem 2) Suppose that Γ is a strongly regular multigraph with parameters (m, n, µ, γ, R) with geometric parameter (r, k, t, c), i.e., (m, n, µ, γ, R) = (r, n, rt, rc, r(k − 1)) with integers r > 3 and t > 1, and real numbers k > 0 and c > 0. If k > (_√8

3r + r + 5)rt, k > (c + 1)t, and r(c + r − 1) 6 (r − 1)t, then Γ is the point graph of a 11

2-design with parameters (r, k, t, c).

We will show µx= r(= m), where µx := number of cliques contains x, for each ver-tex x (Lemma 4.4.11) by showing that the set Vs = {x ∈ V |µx > r} is empty. Toward this goal, we will find an upper bound forP_x∈V sxwhere sx :=

P x∈C

P

C6=C0|C ∩C0|2.

We then derive Lemmas 4.4.2 ∼ 4.4.7 in terms of elementary counting techniques and taking advantage of some inequalities.

Find an upper bound for Pµx

i=1(|Ci| − 1), and in particular its exact value whenever

µx= r (Lemma 4.4.2), and upper bound for the size of maximal cliques (Lemma 4.4.4) and their lower bound when they contain a vertex x with µx = r (Lemma 4.4.3). We then find an upper bound for P_C6=C0_∈M|C ∩ C0|2 in terms of

P

x∈V(µx− r) (Lemma 4.4.5). Moreover, an upper bound for P_x∈V(sx− 3rtk(µx− r)) (Lemma 4.4.6), lower bounds for µxsxin terms of

P_µx

i=1(|Ci| − 1)2 and max{|Ci||i 6 µx} (Lemma 4.4.7 (1,2)) respectively; also lower bound for sx (Lemma 4.4.7 (3)) and for µxsx (Lemma 4.4.7 (4)) for vertex x with µx= r, and µx > r respectively.

We show that there exist at most 2rk vertices x satisfying µx > r (i.e., |Vs| =

vs 6 2rk, Lemma 4.4.8) by studying upper and lower bounds of tx − s0 where

point x with µx > r lies in at least five normal cliques consisting of all points y with

µy > r (Lemma 4.4.9) by a contradictory argument together with some inequalities.

Moreover, we show |C1∩ C2| 6 µ for any distinct maximal cliques C1, C2; and then

show that C − (S_C0_∈NC0) is nonempty for any subfamily N of maximal cliques with

|N| 6 4r and C ∈/ N (Lemma 4.4.10).

We first claim that µx = r is equivalent to the emptiness of the set Ms:= {C|∀x ∈

C, µx > r} by Theorem 4.3.1 and Lemma 4.4.9, we then claim that Ms is empty in terms of the principle of inclusion and exclusion and some inequalities by contradic-tory argument over he conditions |N| 6 4r and the fact that each point x with µx> r lies in at least five cliques.

Suppose that there exists an integer a such that Theorem 4.3.1 (a,b,c) and |C| >

k − aµ for each C ∈ M, and m 6 a 6 2m − 1 are fulfilled.

Let m = r, n = k + r + c − 1 − t, µ = rt, γ = rc, R = r(k − 1). Assume that

k > rt(4a + r + 5), k > (c + 1)t, r > 3 and r(c + r − 1) 6 (r − 1)t.

The relation of Lemma 4.4.2 ∼ Lemma 4.4.11:

Lemma 4.4.5 Lemma 4.4.6

Lemma 4.4.4 Lemma 4.4.7 (2) Lemma 4.4.7 (3) Lemma 4.4.8 Lemma 4.4.7 (1) Lemma 4.4.2 Lemma 4.4.7 (4)

Lemma 4.4.3 Lemma 4.4.9 Lemma 4.4.10

Lemma 4.4.11

Lemma 4.4.2 ([7], Lemma 3.10) l := µx, x ∈ C1, C2, . . ., Cl. Then (1) Pl_i=1(|Ci| − 1) 6 r(k − 1) +1₂γ;

(2) Pl_i=1(|Ci| − 1) = r(k − 1) if l = r.

From Pl_i=1(|Ci| − 1) = r(k − 1), we have:

Lemma 4.4.3 ([7], Lemma 3.12) If µx = r and x ∈ C, then |C| > k − (r − 1)(c + 1).

Lemma 4.4.4 ([7], Lemma 3.11) ∀C ∈ M, |C| < k + c + 1. Lemma 4.4.5 ([7], Lemma 3.13)P_C6=C0_∈M|C ∩C0|2 6 v(r −1)t+3r(c+1) P x∈V(µx− r). Lemma 4.4.6 ([7], Lemma 3.14)P_x∈V(sx− 3rtk(µx− r)) 6 vt(r − 1)(k + c + 1).

Lemma 4.4.7 Let l := µx, x ∈ C1, C2, . . . Cl. Let zi := |Ci| − 1, i = 1, 2, . . . , l, and z := max{zi|i = 1, 2, . . . , l}. Then

(1) l · sx > (n − m)(R + γ) + Rµ − 2(1 + l − m)zγ − P_l i=1zi2 ([7], Lemma 3.15); (2) l · sx> r(k − 1)2+ r(r − 1)tk − r(r − 1)t − rtc − r(c + 1)2− 2γ(k + c)(l − r) − P_l i=1z2i ([7], Lemma 3.16);

(3) sx > (r − 1)tk − (r − 1)t2 for every vertex x satisfying µx = r ([7], Lemma 3.17); and

(4) µxsx > r(r − 1)tk + k(k − 4µ − aµ)(µx− r) for every vertex x satisfying µx > r ([7], Lemma 3.18).

Lemma 4.4.8 ([7], Lemma 3.19) There exist at most 2rk vertices x satisfying µx > r, i.e., vs6 2rk.

Lemma 4.4.9 ([7], Lemma 3.20) Every vertex x satisfying µx > r lies in at least five cliques, which contain only vertices y satisfying µy > r.

Lemma 4.4.10 ([7], Lemma 3.21) (1) |C1∩ C2| 6 µ for C1 6= C2.

(2) C ∈ M, N ⊆ M with |N| 6 4r and C ∈/ N , then there exist x ∈ C and x ∈/

C0 _{∈ N for all C}0_.

Lemma 4.4.11 ([7], Lemma 3.22) µx = r for each vertex x.

5

A Class of Strongly Regular Multigraphs

In this section, we will use the special properties of the definition of alternating form graph to define a symmetric association scheme. From the symmetric association scheme, we have that the alternating form graph is a distance regular graph. At last, we will define a graph Γ which is the induced subgraph of the alternating form graph, and give the multiplicity on the edges, and we will get a class of strongly regular multigraphs.

Definition 5.1 The alternating form graph Alt(n, q) is the simple graph with ver-tex set V = {A|A ∈ Mn×n(GF (q)), A = −AT} and the edge set E = {(A, B)|A, B ∈

V, rank(A − B) = 2}.

Since the alternating form graph Alt(n, q) is defined on the set of all skew-symmetric

n × n matrices over GF (q), the rank of A − B is 2i for any two matrices A and B in Alt(n, q). Let Ri = {(A, B)|A, B ∈ Alt(n, q), rank(A − B) = 2i}, the relation classes

{R0, R1, . . . , Rd}, defined on Alt(n, q) × Alt(n, q), where d = dn₂e; then (1) (Alt(n, q), (Ri)di=0) is a symmetric association scheme.

(2) (Alt(n, q), R1) is a distance regular graph with the intersection numbers {b0, b1, . . . , bd−1; c0, c1, . . . , cd}.

Theorem 5.2 Each maximal cliques of the alternating form graph Alt(n, q) is either

isomorphic to          0 x2 x3 . . . xn −x2 0 0 . . . 0 −x3 0 0 . . . 0 ... ... ... ... ... −xn 0 0 . . . 0          or to             0 x y 0 · · · 0 −x 0 z 0 · · · 0 −y −z 0 0 · · · 0 0 0 0 0 · · · 0 ... ... ... ... ... 0 0 0 0 0 · · · 0            

respec-tively, called typed I and type II respectively.

For studying possible combinatorial geometric structures over Alt(n, q) , we shall study the matrix representations of those maximal cliques of both types containing the zero form. The others may be obtained simply by translation. Without loss of generality, we may assume that V is an inner product space with a fixed orthonormal basis {v1, v2, . . . , vn}. If v ∈ V is a nonzero vector, then < v >=< αv > for all

α ∈ GF∗_{(q), we may assume that v =}P

16j6k−1αjvj+ vk for some k and αj ∈ GF (q). Consider the nest

V0 ⊂ V1 ⊂ V2 ⊂ . . . ⊂ Vn−1 ⊂ Vn(= V )

of subspaces of V , where Vi =< v1, v2, . . . , vi >, 1 6 i 6 n, and in particular V0 is

the trivial subspace of V . Since dimVi is i, Vi has (qi − 1)/(q − 1) one-dimensional subspaces, they are

< X 16j6k−1

αjvj+ vk >

hyperplanes) are

< X 16j6k−1

αjvj + vk>⊥=< vk+1, vk+2, . . . , vn, vj − αjvk|1 6 j 6 k − 1 > .

Proposition 5.3 Show that each of these hyperplanes uniquely determines a maximal clique of type I of the zero form.

Proof: When k = 1, v = v1 and < v >⊥=< v2, v3, . . . , vn>, it follows that its matrix representation is lv(= l∞) = { Ã 0 x2 x3 . . . xn−1 xn −x2 0 0 . . . 0 0 ! |xi ∈ GF (q)}

consisting of those matrices whose first two rows as shown above, their first two columns obtained by skew-symmetry, and zero all other entries. Similar convention is used in the following.

When k > 2, the clique determined by v =P_16j6k−1(αjvj + vk) or its perpendic-ular space is denoted by lv = lα1,α2,...,αk−1 if there is no confusion. Let {e1, e2, . . . , en}

be the standard basis of GF (q)n_{, i.e., e}

i = (0, 0, . . . , 0, 1, 0, 0, . . . , 0) with 1 in the i-th entry, 1 6 i 6 n.

Let

Li = {eTi · x − xT · ei|x ∈ GF (q)n with 0 in its i-th entry}, and,

α = (α1, α2, . . . , αi−1, 1, 0, 0, . . . , 0) ∈ GF (q)n

and Pi(α) denote the matrix obtained from the identity matrix by replacing its i-th row by the vector α. Then the matrix representation of lv = lα1,α2,...,αi−1, with respect

lv = lα1,α2,...,αi−1

= Pi(α)T · Li· Pi(α)

= {Pi(α)T·(ei·x−xT·ei)·Pi(α)|x ∈ GF (q)n with 0 in its i-th entry} = {αT _{· x − x}T _{· α|x ∈ GF (q)}n _{with 0 in its i-th entry}.}

Remark For attenuated space, i.e., Mk×n(GF (q)), those blocks of the zero matrix can be expressed as {αT _{· x|x ∈ GF (q)}n_{} where α = (α1, α2, . . . , α}

i−1, 1, 0, 0, . . . , 0) ∈

GF (q)n _{for all nonzero α ∈ GF (q)}n _{with 1 in its last nonzero entry.}

For each i, there are qi−1 _{vectors of the form}

α = (α1, α2, . . . , αi−1, 1, 0, 0, . . . , 0) ∈ GF (q)n,

and each such vector α uniquely determines a maximal clique lv = lα1,α2,...,αi−1 =

Pi(α)T · Li· Pi(α) of zero form. Hence all maximal cliques of type I of the zero form are obtained in this way. The above observation can be summarized as follows:

Proposition 5.4

(1) Each maximal clique of type I consists of qn−1 _vertices. (2) Each vertex is in exactly q_q−1n−1 maximal cliques of type I.

Proof: The maximal clique of type I consists of qn−1 _{vertices. since all maximal cliques}

of type I is isomorphic to          0 x2 x3 . . . xn −x2 0 0 . . . 0 −x3 0 0 . . . 0 ... ... ... ... ... −xn 0 0 . . . 0         

. And the maximal clique is

uniquely determined by hyperplane H with dim(H) = n − 1. Then the number of hyperplanes is " n n − 1 # q = q_q−1n−1. Q.E.D.

zero form can be expressed as

{αT _{· x − x}T _{· α|x = (x}

1, x2, . . . , xi−1, 0, xi+1, xi+2, . . . , xn), xj ∈ GF (q)n}, for α = (1, 0, 0, . . . , 0) or α = (α1, α2, . . . , αi−1, 1, 0, 0, . . . , 0) where i > 2 and αj ∈

GF (q)n_.

Now, we turn to the intersection properties among those maximal cliques in the distance regular graph Alt(n, q).

Lemma 5.6 Let C1 and C2 be two maximal cliques of type I, then |C1 ∩ C2| = 0

or q. Moreover, if C1∩ C2 6= φ, then C1∩ C2 is isomorphic to

            0 x 0 0 . . . 0 −x 0 0 0 . . . 0 0 0 0 0 . . . 0 0 0 0 0 . . . 0 ... ... ... ... ... ... 0 0 0 0 . . . 0            

and there exists another q − 1 maximal cliques C3, C4, . . ., Cq+1 of type I such that P_q+1

i=1Ci = C1∩ C2.

Proof: Clearly, |C1 ∩ C2| = 0 or q. C1 and C2 are determine by hyperplanes H1 and H2, respectively, with dim(Hi) = n − 1, and dim(H1∩ H2) = n − 2 for i = 1, 2. In

addition to H1, H2, there are another q − 1 hyperplanes containing H1 ∩ H2. They

determine the rest q − 1 maximal cliques of type I are required. Q.E.D.

Let B be the set of all maximal cliques of type I in Alt(n, q). Then π = (An, B, ∈) turns out to be an incidence structure with some intersecting properties. Elements in

B are called blocks of this incidence structure. The following proposition is simply a

Proposition 5.7

(1) Each block B consists of qn−1 _points,

(2) Each point is incident with exactly (qn_{− 1)/(q − 1) blocks,}

(3) Any two distinct blocks are incident with either 0 or q common points,

(4) If B1, B2 ∈ B are distinct and B1∩ B2 6= φ then there are another q − 1 blocks,

say B3, B4, . . ., Bq+1, such that T_q+1

i=1Bi = B1∩ B2 consists of q points.

Definition 5.8 A singular line is the intersection of two distinct maximal cliques.

When we only consider the maximal cliques of type I, we have the constant size of cliques. And if we define the graph with multiedge by mA,B = the number of singular lines containing vertices A and B, then we have mA,B = 0, 1 or q + 1.

Consider the definition of strongly regular multigraph, γ = P_y∈V mx,y(mx,y − 1). For each A and B lie in the same maximal cliques, there are exactly 2 entries of B different from those of A. So we have γ =P_B(q + 1)q is the constant.

Definition 5.9 For a fixed matrix M0

(n−2)×(n−2)(GF (q)) with odd q, let Γ be a

multi-graph with defined on (V, E) with {M| M is a skew-symmetric matrix over GF (q) satisfying M =

Ã

X Y

−YT _M0 !

} as the vertex set, and for any two vertices A, B, A ∼ B if and only if rank(A − B) = 2. Moreover, let mA,B be the number of singular line containing vertices A and B.

Theorem 5.10 Γ is a strongly regular multigraph with parameters

and Γ is the collinearity graph of a 11

2-design with parameters

(r, k, t, c) = (q + 1, qn−1_{, q}2_{, q(q − 1)).}

Proof:

For a fixed vertex A ∈ V ,

(1) m0 _{= q + 1 is the number of maximal cliques contains A;}

(2) there exists exactly q + 1 cliques containing A and B for adjacent vertices A and

B, and each maximal clique has size qn−1_{; thus R}0 _{= (q + 1)(q}n−1_{− 1).} (3) γ0 ₌P

B∼AmA,B(mA,B− 1) = P

B∼A(q + 1)q = (q − 1)q(q + 1) = q(q2− 1), since

mA,B = 1 or 0 whenever B does not lie in the q +1 maximal cliques which contains

A.

(4) since m0_(m0_{− n}0_{) + µ}0 _{= R}0 _{+ γ}0_{, then (q + 1)(n}0_{− (q + 1)) + µ}0 _{=(q + 1)(q}n−1₋ 1) + q(q2 _{− 1), it follows that µ}0 _{= (q + 1)(q}n−1_{− n}0_{+ q}2_).

(5) since µ0_{v = (R}0 _{+ m}0_)(R0 _{+ m}0 _{− n}0_{), then µ}0_q2n−3 _{= ((q + 1)(q}n−1 _{− 1)) + (q +} 1)((q + 1)(qn−1_{− 1) + (q + 1) − n}0_{). It follows that µ}0_qn−2 _{= (q + 1)((q + 1)q}n−1_{− n}0_); combine (4) and (5), we have µ0_qn−2 _{= q}n−2_{(q + 1)(q}n−1_{− n}0 _{+ q}2_{). Hence n}0 _{= q}n−1 and µ0 _{= (q + 1)(q}n−1_{− n}0 _{+ q}2_{) = q}2_{(q + 1).}

We then have (r, k, t, c) = (q + 1, qn−1_{, q}2_{, q(q − 1)) as required. Q.E.D.}

The combinatorial interpretations of the parameters (r, k, t, c) of the 11

2-design

un-der consiun-deration are given below:

(1) r = number of cliques contains a fixed vertex = q + 1. (2) k = the clique size = qn−1_.

(3) Fixed y ∈/ B where y is a point and B is a block, then t =P_x∈Bmx,y = q2. (4) Fixed y ∈ B, mx,y− 1 = q or 0 by x ∈ B and

a. mx,y = 1 if x does not lie in the q + 1 blocks containing y, b. mx,y = q + 1 if x lies in the q + 1 blocks containing y,