圖與重邊圖結合各式的設計之研究

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國 立 交 通 大 學

應用數學系

碩士論文

圖與重邊圖結合各式的設計之研究

A Study of Graphs and Multigraphs

Associated with Various Designs

研究生 : 吳介友

指導老師 : 黃大原 教授

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圖與重邊圖結合各式的設計之研究

A Study of Graphs and Multigraphs

Associated with Various Designs

研 究 生:吳介友 Student:

Chieh-Yu Wu

指導教授:黃大原 Advisor:

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 the Requirements

for the Degree of

Master

in

Applied Mathematics

June 2006

Hsinchu, Taiwan, Republic of China

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謝 誌

初到交大,有著些許的不習慣,所幸老師及同學們都很好,助我跨越這段過 渡時期。感謝指導教授 黃大原老師,從老師作研究的嚴謹態度及追根究底的精 神,使我受益良多,謝謝您這兩年的指導。 也要感謝三位組合組的老師: 陳秋媛老師,如同母親般提供生活上的照顧及鼓勵。 翁志文老師,不厭其煩幫我釐清數學觀念和有耐心地陪我解決問題。 傅恆霖老師,提供各方面資訊,使我在離散數學這塊領域的觸角更加廣泛。 再則謝謝同學們, 應數真男人-余國安,RE-張雁婷,牧師-卜文強,肌肉-張澍仁, 收斂王子-李張圳,理查-陳柏澍,貓貓-曾妙玲 感謝你們陪我一起度過研究 所的兩年時光。在我有困難時,適時地伸出援手;一起出遊、聚餐的快樂景象, 都深深地記憶在我腦海深處,永不忘懷。 豪哥-梁育豪,麻將-陳冠羽,老謝-謝俊鴻,金曲歌王-陳宜廷,育慈,筱凡 遇到關於分析方面的問題時,感謝有你們的幫助;在生活及 Tex 等…各方面都受 到你們許多的協助。 同時感謝學長姐們、威雄、兆涵、敏筠、鈺傑、佩純、偉帆、子鴻受到你們許多 的照顧 最後感謝我的家人,爸爸、媽媽、妹妹, 因為有你們的支持,讓我可以無牽無掛地專心讀書,是使我撐下去的最大動力, 我的親愛家人們。

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圖和重邊圖結合各式的設計之研究

研 究 生:吳介友

指導老師:黃大原 教授

國 立 交 通 大 學

應 用 數 學 系

Bose 首先提出強正則重邊圖的概念,接著 Neumair 和 Metsch 利用強正則重邊圖的 概念進一步地解決準剩餘 2-設計的問題。近來,不完全幾何設計的概念被 van Dam 和 Spence 使用在具有 2 個奇異值的組合設計。我們將 Neumair 和 Metsch 兩篇論文 中的定義與結果做整理,並以統一形式呈現在此論文裡,進而舉出一些 2-設計及其 對應的強正則重邊圖。藉由這些圖,研究具有 3 或 4 個相異特徵值的連通正則圖之 特性。

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A Study of Graphs and Multigraphs Associated

with Various Designs

Student: Chieh-Yu Wu

Advisor: Tayuan Huang

Department of Applied Mathematics National Chiao Tung University

Hsinchu, Taiwan 30050

Abstract

The notion of strongly regular multigraphs was first introduced by R. C. Bose, followed by Neumaier for characterizing quasi-residual 2-designs, and fur-ther by Metsch for embeddings of residual 2-designs. Recently, the notion of partial geometric designs was also used by van Dam and Spence over com-binatorial designs with two singular values. The basic definitions and most results regarding strongly regular multigraphs and partial geometric designs covered in the works of Neumaier and Metsch are given in a unified way in this thesis. The associated multigraphs or graphs of 2-designs are then studied, fol-lowed by a few examples of 2-designs and their corresponding strongly regular multigraphs. Motivated by these graphs, connected regular graphs with 3 or 4 distinct eigenvalues are also studied.

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Contents

Abstract (in Chinese) i

Abstract (in English) ii

Contents iii

1 Introduction 1

2 Basic Definitions and preliminary 2 2.1 Graphs and multigraphs . . . 2 2.2 Designs . . . 12 2.3 11

2 -designs(or called partial geometric designs) . . . 13

3 The graphs and multigraphs associated with some designs 23

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1

Introduction

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

2-designs, Neumaier gave an equivalent definition of strongly regular multigraphs in

an elegant and self-contained way [2]. Metsch continued the study of embeddings of residual 2-designs within the framework of strongly regular multigraphs. However, the notations used by them are quite different.

Neumaier showed that the block multigraph of a 2-design of order n is a strongly regular multigraph, together with a partial converse with some constraints over its parameters. Its proof involves 2-designs and its variations, called 11

2 - designs (or called partial geometric designs), or weak 11

2 - designs. Neumaier showed also that a strongly regular multigraph under some numerical constraints is the point multigraph of a unique 11

2 - design. Recently, the notion of partial geometric designs was also used by van Dam and Spence [3, 4] over combinatorial designs with two singular values.

Though strongly regular multigraphs and partial geometric designs are the com-mon themes covered in [7, 6], the notations used by Neumaier and Metsch are quite different. We expect that these notions will keep playing essential roles in the fu-ture. The basic definitions and most results regarding strongly regular multigraphs and partial geometric designs covered in [7, 6] are given in a unified way in section 2. The block multigraphs or associated block graphs, and the point multigraphs of 2-designs are studied in section 3, followed by a few examples of 2-designs and their corresponding strongly regular multigraphs. Motivated by these graphs, connected regular graphs with 3 or 4 distinct eigenvalues are studied in section 4.

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2

Basic Definitions and preliminary

2.1

Graphs and multigraphs

Definition 2.1. A graph is a triple consisting of a vertex set V (Γ), an edge set E (Γ), and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints. Two nonloops are parallel when they have the same ends; graphs that contain them are called multigraphs. Equivalently, a multigraph Γ consists of a nonempty set V of vertices and a multiset E of edges.

For a multigraph Γ and x, y∈V = V (Γ), let mxx:=0 , and mxy:=the number of

edges joining x and y.

Definition 2.2. The eigenvalues of an adjacency matrix A(Γ) = A of a connected graph are called eigenvalues of the graph Γ, denoted by Spec(Γ) = (θ0m01m12m2,. . . ,θsms)

with θ0 > θ1 > θ2 > . . . > θs and mi is the multiplicity of the eigenvalue θi.

Definition 2.3. For a connected k -regular graph of diameter d with Spec(Γ) = (k1 1m12m2,. . . ,θsms), then h(x) = Y 1≤i≤s (x − θi) Y 1≤i≤s (k − θi)

is called the Hoffman polynomial of the graph Γ.

Definition 2.4. A simple graph Γ is strongly regular if there are parameters k, λ, µ such that Γ is k -regular, every adjacent pair of vertices have λ common neighbors, and every nonadjacent pair of vertices have mu common neighbors, denoted by SR(v,

k, λ, µ), where v is the number of vertices in Γ.

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For a connected graph Γ with an adjacency matrix A, then Γ is a strongly regular graph if and only if

AJ = kJ, A2 = kI + λA + µ(J − I − A) = (λ − µ)A + (k − µ)I + µJ.

A multigraph version of strongly regular graphs is defined by Bose and Shrikhande 1973; similar to strongly regular graphs, the matrix expressions in terms of their adjacency matrices A = [mi,j] for strongly regular multigraphs are given.

Definition 2.5. [1, 6] A multigraph Γ is called a strongly regular multigraph (SR multigraph) with parameters (m, n, µ, γ, R) for real numbers m, n, µ, γ, R with

n > 0, if 1. X x∈V max = R for each a ∈ V ; 2. X x∈V

maxmbx = (n − 2m)mab+ m(n − m)δab+ µ, where δaa = 1 and δab = 0 for

a 6= b.

3. X

x∈V

max(max− 1) = γ for each x ∈ V .

4. vµ = (R + m)(R + m − n).

Indeed, condition 4 is a consequence of the fact that X x∈V (max)2 = X x∈V max+ X x∈V

max(max− 1) and the conditions 1 3 above.

Note that the parameters γ =X

y∈V

mxy(mxy − 1), R =

X

y∈V

mxy for each vertex x

were defined explicitly, though m, n, µ and are not. The parameter

γ = X

y∈V

mxy(mxy − 1) measures the derivations of the multigraphs from graphs; if

γ = X

y∈V

mxy(mxy − 1) = 0, then either mxy = 0 or mxy = 1 for x, y ∈ V . The

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case γ = 0. Some subsets of vertices including claws, maximal cliques play essential roles in the study of their geometric structures. The parameter m is the constant number of maximal cliques containing a fix vertex in the study of maximal claws under some numerical constraints.

Proposition 2.1. [6] An SR multigraph Γ with parameters (m, n, µ, γ, R) with

γ = 0 is an SR graph with parameters

(m, n, µ, γ, R) = (R(R − n + 2m − µ − 1)

µ + R + 1, R, n − 2m + µ, µ)

Proof : Let A = [mxy], it suffices to show that mxy ≥ 0 for all x, y ∈ V (Γ).

Consider the (a, b) entry of a, b ∈ V (Γ): 1. X x∈V (Γ) = R = k for each a ∈ V (Γ). 2. X x maxmbx= (n − 2m)mab+ m(n − m)δab+ µ, when a = b X x maxmax= X x max = m(n − m) + µ = R = k;

when a, b are adjacent, X

x

maxmbx= (n − 2m)mab+ µ = (n − 2m) + µ = λ;

when a, b are nonadjacent, X

x

maxmbx= µ.

Since k(k − λ − 1) = µ(v − k − 1) for SR graphs, we have

v = k(k − λ − 1)

µ + k + 1 =

R(R − n + 2m − µ − 1)

µ + R + 1.

The following lemma will be used in the proof of an inequality below.

Lemma 2.2. [7] Let A be an integral symmetric matrix with zero diagonal satisfying

AJ = RJ, A2 = (n − 2m)A + m(n − m)I + µJ. If

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where γ = m(n − m) + µ − R, then A is the adjacency matrix of a SR multigraph (m, n, µ, γ, R).

Proof : Since A2 = (n − 2m)A + m(n − m)I + µJ, then

X x maxmbx = (n − 2m)mab+ m(n − m)δab+ µ, X x max(max− 1) = γ = m(n − m) + µ − R. For a 6= b, we have: 2mab(mab− 1) ≤ X x (max+ mbx)(max+ mbx− 1) =X x max(max− 1) + 2 X x max+ mbx− X x mbx(mbx− 1) = γ + 2((n − 2m)mab+ µ) + γ. it follows that max(max− 1) ≤ ((n − 2m)mab+ µ) + γ, and (mab+ 1)(mab+ 2) ≤ (n − 2m + 4)mab+ µ + γ + 2 ≥ 0. Since n ≥ max{2m − 4, 2m − 1 + µ + γ}, then (n − 2m + 4)mab+ µ + γ + 2 ≥ (µ + γ + 3)mab+ µ + γ + 2 ≥ 0, so

mab ≥ 0. Hence A is the adjacency matrix of an SR multigraph (m, n, µ, γ, R).

Some other combinatorial interpretations for the parameters can be found in the following lemma.

Lemma 2.3. [7] For an SR multigraph (m, n, µ, γ, R)

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2. m ≥ 1, with equality if and only if it is the disjoint union of complete graphs.

3. If there are nonadjacent vertices, then n ≥ m.

4. µ ≥ (R + m)(m − n), with equality if and only if mab = m − n for all a 6= b.

5. µγ ≤ (n − 2m + µ)(m(n − m) + µ), with equality if Γ contains no triangles.

6. µ ≥ 2m − n. 7. If n ≤ 2m + 4, then γ < 2m(n − m) + n − 2m − 1 + µ. Proof : 1. 2mab2 = mab2+ mab2 X x=a or b (max− mbx)2+ X x6=a or b (max− mbx)2 =X x max2− 2 X x maxmbx+ X x mbx2 = 2m(n − m) − 2(n − 2m)mab ⇒ mab2 + (n − 2m)mab− m(n − m) ≤ 0 ⇒ (mab− m)(mab+ (n − m)) ≤ 0. Then m − n ≤ mab ≤ m.

2. For adjacent a, b, 1 ≤ mab ≤ m ⇒ 1 ≤ m by 1 above.

If mab = 1, since 2mab2 X x=a or b (max− mbx)2+ X x6=a or b (max− mbx)2, then mab = 1 =

mba and max = mbx, i.e. a and b are joined to exactly the same points. This implies

that it is the disjoint union of complete graphs.

3. For nonadjacent a, b, then m − n ≤ mab = 0 and hance m ≤ n by (1).

4. γ = m(n − m) + µ − R ≥ 0

⇒ µ ≥ R + m(m − n) ≥ R(m − n) + m(m − n) (since m − n ≤ 0) ⇒ µ ≥ (R + m)(m − n)

If mab = m − n for distinct a, b, then

X

x

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⇒ ((R + m)(R + m − n) µ − 2)(m − n) 2 − (n − 2m)(m − n) = µ (since vµ = (R + m)(R + m − n)) ⇒ µ2+ (n − 2m)(m − n)µ − (m − n)2((R + m)(R + m − n) − 2µ) = 0 ⇒ [µ − (R + m)(m − n)] · [µ + (R + m − n)(m − n)] = 0 µ = (R + m)(m − n) or µ = −(R + m − n)(m − n).

Since µ ≥ (R + m)(m − n), we assume µ = (R + m)(m − n) + s for some s ≥ 0,

0 ≤X x6=a (max− R v − 1) 2 =X x6=a max2− 2 R v − 1 X x6=a max+ X x6=a ( R v − 1) 2 = m(n − m) + µ − R 2 v − 1 = s − R(m − n) − R 2 v − 1 = s(nR − s) µ(v − 1) When s = 0, then max = R v − 1 = R R m − n

= m − n for each x 6= a. 5. For a fixed

point a, the number of triangles containing a is

X x X y maxmxymya = X x max( X y mxymya) =X x max((n − 2m)max+ µ) = (n − 2m)(m(n − m) + µ) + µ(m(n − m) + µ − γ) = (n − 2m + µ)(m(n − m) + µ) − γµ ≥ 0, it follows that γµ ≤ (n − 2m + µ)(m(n − m) + µ). When γµ = (n − 2m + µ)(m(n − m) + µ), then X x X y maxmxymya = 0, i.e. the number of triangles is 0.

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and (n − 2m + µ) ≥ 0, hence µ ≥ 2m − n.

7. Let Γ be an SR multigraph (m, n, µ, γ, R) with an adjacency matrix M, then

MJ = RJ . . . (∗) M2 = (n − 2m)M + m(n − m)I + µJ . . . (∗∗) Let M0 = −M,then M0J = −MJ = −RJ = R0J, and (M0)2 = (−M)2 = M2. Hence, (n0− 2m0)(−M) + m0(n0− m0)I + µ0J = (n − 2m)M + m(n − m)I + µJ.

Compare the coefficients of the above two equations, we have

m0 = n − m, n0 = n, µ0 = µ, R0 = R,

γ0 = m0(n0− m0) + µ0− R0 = 2m(n − m) + 2µ − γ.

By lemma 2.2 [7], it follows that if n ≤ 2m + 4, then γ < 2m(n − m) + n − 2m − 1 + µ, as required.

Similar to SR graphs, the matrix expressions in terms of their adjacency matrices

A = [mij] for strongly regular multigraphs are given below.

Lemma 2.4. [7] Let A be an adjacency matrix of a multigraph Γ of order v, then the following are equivalent:

1. Γ is an SR multigraph (m, n, µ, γ, R),

2. AJ = RJ and A2 = (n − 2m)A + m(n − m)I + µJ

= (m(n − m) + µ)I + (n − 2m + µ)A + µ(J − I − A) for some real numbers R, m, n, µ with n > 0.

Moreover, v = (R + m)(R + m − n)

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Proof : Let A be an adjacency matrix of an SR multigraph Γ with vertex set X.

1. Since X

x

max= R for each a ∈ X, hence AJ = RJ.

2. For a, b ∈ X, X x maxmbx= (n − 2m)mab+ m(n − m)δab+ µ gives A2 = (n − 2m)A + m(n − m)I + µJ . . . (∗)

Multiplying both sides of (*) by J, and J2 = vJ with v = |X|, then

A2J = (n − 2m)AJ + m(n − m)IJ + µJ2, and

R2J = R(n − 2m)J + m(n − m)J + µvJ, It follows that R2 = R(n − 2m) + m(n − m) + µv, and µv = R2 − (n − 2m) − m(n − m) = (R + m)(R + m − n), hence v = (R + m)(R + m − n) µ . Conversely, X x max(max− 1) = X x max2 X x max = m(n − m) + µ − R is a

constant, denoted by γ, and hence R + γ = m(n − m) + µ.

Lemma 2.5. [7, 6] Suppose Γ is an SR multigraph (m, n, µ, γ, R), then 1. There are unique k > 1, r > 0 and t, c ≥ 0 such that

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

and (r, k, t, c) is called the geometric parameters of this multigraph.

2. The number of vertices of an SR graph with parameters (m, n, µ, γ, R), or with geometric parameter (r, k, t, c), is

v = (R + m)(R + m − n)/µ = r((r − 1)(k − 1) + t − c)/t.

The parameter µ in SR multigraph(m, n, µ, γ, R) is identical with that of µ in SR graph(v, k, λ, µ) in case γ = 0. Some subsets of vertices including claws, maximal

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cliques play essential roles in the study of their geometric structures. The parameter m is the constant number of maximal cliques containing a fix vertex in the study of

maximal claws under some numerical constraints.

A clique of a multigraph is a set of pairwise adjacent points; a clique which cannot be extended to a larger clique is called maximal clique. In an SR multigraph (m, n, µ, γ, R), motivated by the following lemma, a maximal clique C with |C| > (n/2) + µ + 1 − m is called a grand clique.

Lemma 2.6. [7] In an SR multigraph (m, n, µ, γ, R), an edge of multiplicity1 is in at most one grand clique.

Proof : Let ab be an edge of multiplicity 1 contained in two distinct grand cliques C and C0. Since C and C0 are maximal, there is x ∈ C0 such that C ∪ {x} is not a

clique, and hence there is y ∈ C with mxy = 0.

1. The points z ∈ C∩C0are adjacent to x and y and hence |C∩C0| ≤X z

mxzmyz = µ.

2. The points z ∈ |C ∪ C0| − {a, b} are adjacent to both a and b, whence

|C ∪ C0| − 2 ≤X z

mazmbz = n − 2m + µ.

Hence |C| + |C0| = |C ∩ C0| + |C ∪ C0| ≤ n + 2(µ + 1 − m), this contradicts the fact

that both |C| and |C0| are grand cliques.

Theorem 2.1. [7] If C is a clique of an SR multigraph(m, n, µ, γ, R) with µ > 0, then

|C|(R + m − µ) ≤ (n + 1 − m)(R + m).

Equality holds if and only if

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2. for x /∈ C, there are a constant number α of edges containing x and intersecting C ; in this case, α = |C| + m − 1 − n.

Proof : Let C be a clique with |C| = c points. Define αx =

X

x∈C

max. Then, for

x /∈ C, α is the number of edges containing x and intersecting C. We compute the

expression N(α) =X x /∈C (αx− α)2+ X x∈C (αx− α + m − n)(αx− α + m). Since 1. X x 1 = v, 2. X x αx = X a∈C X x max = cR, 3. X x αx2 = X a, b∈C X x maxmbx = (n − 2m)X x∈C αx+ m(n − m)c + µc2, Whence N(α) =X x (αx− α)2+ (2m − n) X x∈C αx+ (m(m − n) − α(2m − n))|C| = µc2− cα(2R + 2m − n) + α2v = µ−1(cµ − α(R + m))(cµ − α(R + m − n)).

In particular, for α = cµ/(R + m), N(α) = 0, and we conclude from

N(α) =X x /∈C (αx− α)2+ X x∈C (αx− α + m − n)(αx− α + m)

that αx ≤ α + n − m for all x ∈ C since otherwise N(α) would be strictly positive.

But, for x ∈ C, αx =

X

a∈C

max ≥ c − 1 since C is a clique. Hence c − 1 ≤ α + n − m

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If equality holds then αx = c − 1 = α + n − m for all x ∈ C. Hence, C contains

only edges of multiplicity 1. Moreover, N(α) = 0 implies that αx = α = c + m − 1 − n

for all x /∈ C.

Conversely, if C contains only edges of multiplicity 1, and αx = α0for all x /∈ C(for

some α0) then α

x= c − 1 for all x ∈ C, and we obtain from 1∼3.

2.2

Designs

Definition 2.6. Let X be a set of v points and B ⊆¡Xk¢ such that any two elements of X lie in exactly λ blocks, then (X, B) is called 2-(v, k, λ)design

Definition 2.7. A 2-(v, k, λ) design has exactly b blocks, and every point occurs in exactly r blocks.

1. A 2-(v, k, λ)design is called a symmetric design if b = v(or, equivalently, r = k or λ(v − 1) = k2− k).

2. A 2-(v, k, λ)design is called a quasi symmetric if the cardinality of the inter-section of two distinct blocks takes only two distinct values.

Note that for a 2-(v, k, λ) design,

r = λ(v − 1)/(k − 1), b = λv(v − 1)/k(k − 1) and v = k + n(k − 1)

λ where n = r − λ is the order, and v = 1 + k(k − 1)

λ for symmetric designs.

Definition 2.8. [7]

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

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2. The point multigraph(collinearity graph) of a 2-design π = (X, β) is the multi-graph defined over the set of points, and two distinct vertices (points) x, y are connected by mxy edges if they are contained in mxy blocks.

3. The block graph of a quasi-symmetric 2-design with sizes x, y of intersections of blocks is defined over the set of blocks, and two distinct vertices (blocks) A, B are adjacent if and only if their intersection has cardinality y.

Theorem 2.2. [7] The block multigraph of a 2-(v, k, λ) design of order n = r − λ is an SR multigraph (m, n, µ, γ, R) with

(m, n, µ, γ, R) = (k, n, k2λ, 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 2.3. [7] Every SR multigraph with parameters

(m, n, µ, γ, R) = (k, n, k2λ, 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)2λ, 2(k − 1)(k2λ + kλ − 2λ + 1),1

2(k

2− 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 .

2.3

1

1

2

-designs(or called partial geometric designs)

It is well known that the block graphs of quasi-symmetric 2-designs are strongly regular. This leads to the question whether strongly regular multigraphs can be associated with some designs of various types? The notion of partial geometric design with parameters (r, k, t, c) was introduced as a generalization of a partial geometry

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with parameters (r, k, t)(with c = 0 above). As a generalization of strongly regular graphs(SR graph), a partial geometric design with parameters (r, k, t, c) gives rise in a natural manner to a strongly regular multigraph(SR multigraph) whose parameters depend on r, k, t and c.

The notion of 11

2 -designs(called partial geometric designs by R.C. Bose 1976). Note that 2-designs, transversal designs, semiregular partially balanced incomplete block designs, partial geometries, and polar spaces are examples of 11

2 -designs. The

block multigraphs of 11

2 -designs, and dually, the point multigraphs of weak 1 1

2 -designs still are strongly regular, and by investigating closely the properties of cliques and claws in a multigraph, general characterization theorems which specialize to Theorem 2.3([7]). 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 SR multigraph and 11

2 -design.

Definition 2.9. [7] An incidence structure with an incidence matrix A is

1. a weak 2-design if AJ = rJ, AAT = nI + λJ and

2. a weak 11

2 -design if AJ = rJ, and AA

TA = nA + λJA.

A class of incidence structure lies between 1-designs (regular) and 2-designs is considered. For an incidence structures, let

mxx = 0, and

mxy= the number of blocks containing points x and y. ([6])

Definition 2.10. [7, 6] A 11

2 -dsign (or called partial geometric design) with param-eter (r, k, t, c) is an incidence structure I = (X, B) such that

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2. each block consists of k points in X ;

3. for a point x and a block B

a. t =X y∈B mxy ≥ 1 is a constant if x /∈ B; b. c = X y∈B−x (mxy− 1) is a constant if x ∈ B.

A partial geometry is a partial geometry design with parameters (r, k, t, 0) with

c = 0, i.e., a semilinear incidence structure such that

1. each point lies on r blocks of B;

2. each block consists of k points in X ;

3. for a point x and a block b with x /∈ B, there are exactly t blocks contain x

meeting b.

Let A be the incidence matrix of the incidence structure under consideration, and (x, B) is a pair of point and block, let c = X

y∈B−x (mxy− 1) for x ∈ B, and t = X y∈B mxy for x /∈ B. If x /∈ B, then AATA(x, B) = X all C X all y

A(x, C)A(y, C)A(y, B)

=X y∈B A(x, C)AT(C, y) =X y∈B mxy (say t). If x ∈ B, then

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AATA(x, B) = X

all C

X

all y

A(x, C)A(y, C)A(y, B)

= 1 + (k − 1) + (r − 1) + X y∈B−x (mxy − 1) = r + k − 1 + c. Remark[6]: If x /∈ B, then α(x, B) = X C6=B X y6=x

A(x, C)AT(C, y)A(y, B) = X y∈B mxy = α (i.e., t); If x ∈ B, then α(x, B) = X C6=B X y6=x

A(x, C)AT(C, y)A(y, B) = X y∈B−x

(mxy−1) = n+α−(r +k −1) = β

(i.e., c).

Lemma 2.7. [7, 6] For a binary matrix A, the following are equivalent:

1. A is the incidence matrix of a 11

2 -design with parameters (r, k, t, c)and t ≥ 1. 2. AJ = rJ, JA = kJ and

AATA = (r + k − 1 + c)A + t(J − A) = (r + k − 1 + c − t)A + tJ(i.e., nA + αJ

in[7]) for some integers r, k, t, c with t ≥ 1.

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adjacency matrices as following: AJ = kJ, A2 = kI + λA + µ(J − I − A) = (λ − µ)A + (k − µ)I + µJ. AJ = RJ, A2 = (n − 2m)A + m(n − m)I + µJ

= (m(n − m) + µ)I + (n − 2m + µ)A + µ(J − I − A).

AJ = rJ, JA = kJ,

AATA = (r + k − 1 + c)A + t(J − A) = (r + k − 1 + c − t)A + tJ.

Lemma 2.8. [7]

1. 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)).

2. Each 11

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

(v, λ) = (1 + r(k − 1)

λ , t + 1 − c − k).

Proof : 1. For a 2-(v, k, λ) design (X, B), each block consists of k points in X

and each points lies on r = λ(v − 1)

k − 1 blocks of B. For a point x and a block b,

if x /∈ b, then t =X y∈b mxy = k · λ ≥ 1; if x ∈ b, then c = X y∈b−x (mxy− 1) = ( X y∈b−x mxy) − (k − 1) = (k − 1)λ − (k − 1) = (k − 1)(λ − 1).

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Hence it is a 11

2 -design with parameters (r, k, t, c) = (λ(v − 1)

k − 1 , k, kλ, (k − 1)(λ − 1)).

2. Let A be an incidence matrix of a 11

2 -design with parameters (r, k, t, c) satisfying (t+1−c−k)k = t. To show AAT = kI +λ(J −I), consider X = AAT−kI −λ(J −I),

then show that X2 is the zero matrix, and hence X = 0 as required.

Theorem 2.4. [6] The collinearity graph of a partial geometry with parameters (r, k, t, c = 0) is an SR graph (v, K, λ, µ) with (v, K, λ, µ) = (r((r − 1)(k − 1) t + 1), r(k − 1), (k − 2) + (r − 1)(t − 1), rt). Proof : v = r((r − 1)(k − 1) + t − c) t = r((r − 1)(k − 1) + t) t = r · ( (r − 1)(k − 1) t + 1), K = r(k − 1) since each point lies on r blocks of B.

If x, y are in the same block, there are (k − 2) points in the block containing x, y. Fix

x, there are (r − 1) blocks containing x but not containing y. Since y is not in those

(r − 1) blocks, then X

y∈b−x

mxy = t − 1 for each b of those (r − 1) blocks. Hence, there

are λ = (k − 2) + (r − 1) · (t − 1) points in the block containing x and y . Let x, y be in the different block, there are r blocks containing x but not y, then X

y∈b

mxy = t

for each b of those r blocks. There are µ = r · t points in the same blocks containing

x and y.

Theorem 2.5. [6] The collinearity graph of a 11

2 -design with parameters (r, k, t, c) is an SR multigraph with parameters

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

Proof : Let A be an incidence matrix of a 11

2 -design with parameters (r, k, t, c), then M = AAT − rI is an adjacency matrix of the cor responding collinearity graph.

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Since AJ = rJ, JA = kJ, and AATA = (r + k − 1 + c − t)A + tJ(i.e., nA + tJ),

We have

MJ = AATJ − rIJ = rkJ − rJ = r(k − 1)J,

AATAAT = (AAT)2 = (r + k − 1 + c − t)AAT + tJAT = nAAT + trJ

⇒ (M + rI)2 = n(M + rI) + trJ

⇒ M2 = (n − 2r)M + r(n − r)I + trJ.

If M is an adjacency matrix of an SR multigraph with parameters (m, n, µ, γ, R), then

MJ = RJ,

M2 = (n − 2m)M + m(n − m)I + µJ.

Compare the coefficients, then

m = r, n = n = (r + k + c − 1 − t), µ = rt,

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

The above lemma shows that the collinearity graph of a 11

2 -design is an SR multi-graph. Following this trend, we are interested in those strongly regular multigraphs which are the collinearity graph of 11

2 -designs? Theorem 2.6 provides sufficient numerical constrains to guarantee the uniqueness of such 11

2 -designs. There is no example of SR multigraphs meeting those numerical constraints found in the papers of Bose [1], Neumaier and Metsch [6]. A class of SR multigraphs associated with the distance regular graphs Alt(n, q) was considered by Huang [5].

Theorem 2.6. [7] If Γ is an SR multigraph (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}

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then Γ is the point multigraph of a unique 11

2 -design, with parameters (r, k, t, c) = (m,R m + 1, µ m, γ m).

The above bound

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} was simproved by Metsch as shown below.

Theorem 2.7. [6]

Suppose that Γ is an SR multigraph whose parameters (m, n, µ, γ, R) = (r, k + r + c − 1 − t, 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 ≈ 5, 6r

2t, k > (c + 1)t, and r(c + r − 1) ≤ (r − 1)t,

then Γ is the collinearity graph of a 11

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

A construction method for cliques in multigraphs was proposed by Metsch [6], this method generalized the ideas used in improving the well-known completion theorem for nets of Bruck. The bound for k in the above 2 improves previous bounds given by Bose et al. [1] and Neumaier [7], however note that the condition r(c+r−1) ≤ (r−1)t did not occur in [7].

Corollary 2.1. [6] Suppose the parameters (m, n, µ, γ, R) of an SR multigraph can be written in the form

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

for some integers k ≥ 3, r, and λ. If

r > (√8

3k + k + 5)k

2λ ≈ 5, 6K3λ, and r > k(k − 1)λ2− k(k − 2)λ,

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Corollary 2.2. [7] Two distinct blocks A and B of a 2-(v, k, λ) design intersect in at least k − r + λ points.

Theorem 2.8. [1, 6] A quasi-residual 2-(w, n, λ) design B is embeddable iff the following three conditions are satisfied:

1. Any distinct blocks A and B intersect µAB ≤ λ points,

2. The multigraph Γ on the blocks, with mAB = λ − µAB edges between A and B,

is a strongly regular multigraph SR(m, n, µ, γ, R) , where

m = λ, n, µ = λ2(λ − 1), γ = λ(λ − 1)(λ − 2), R = λ(n + λ − 2),

3. Γ is isomorphic to the block multigraph of a 2-(n + λ, λ, λ − 1) design B’.

Theorem 2.9. [7] Let B be a quasi-residual 2-(w, n, λ) design with

n ≥ 2λ3− 4λ2+ 4λ − 1.

Then two distinct blocks intersect in at most λ points, and property 2 of Theorem 2.7 holds.

Proof : Since B is quasi-residual, r = n + λ, b = r(r − 1)/λ. Hence, the incidence

matrix A of B satisfies AJ = nJ, JA = (n + λ)J, AAT = nI + λJ. By

straight-forward calculation, the matrix M = (n − λ)I + λJ − AAT satisfies MJ = RJ,

M2 = (n − 2m)M + m(n − m)I + µJ, n > 0 with m = λ, n, µ = λ2(λ − 1),

R = λ(n + λ − 2). Hence, with γ = λ(λ − 1)(λ − 2), Lemma 2.2 applies. Therefore M is the adjacency matrix of a SR(m, n, µ, γ, R), i.e., 2 of Theorem 2.7 holds. In

particular, the offdiagonal entries λ − µAB of M are nonnegative, i.e., two distinct

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The next two results are preliminary conditions for an SR multigraph to be the point multigraph of a weak 11

2 -design, respective a 1 1

2 -design.

Theorem 2.10. [7] An SR multigraph(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 mab is in exactly mab

cliques of P. In this case the blocks are the cliques of P, and the weak 11

2 -design has parameters (v, m, r, λ) with

(v, m, r, λ) = ((R + m)(R + m − n)

µ , n, m,

µ R + m).

Proof : Let g be an SR(m, n, µ, γ, R). If g is the point multigraph of a weak 11

2 -design B then the blocks of B are cliques in g, andP=B satisfies the Conditions of the theorem.

Conversely, if P is a collection of cliques with the stated properties, then define a design B with Pas set of blocks and natural incidence. If A is the incidence matrix of B, then the assumed properties can be stated in terms of A and the adjacency matrix M of g as AJ = mJ, AAT = M + mI.

With λ = µ/(R + m), the property that g is a SR(m, n, µ, γ, R) means MJ = RJ, (M +mI)(M +(m−n)I −λJ) = 0. Hence X = (AAT−nI −λJ)A satisfies XXT = 0,

whence X = 0. Therefore, AATA = nA + λJA, and by vµ = (R + m)(R + m − n),

B is a weak 11

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3

The graphs and multigraphs associated with some

designs

Theorem 3.1. The block graph of a symmetric design is the complete graph Kb, and

the adjacency matrix of the block multigarph of a symmetric design is λ(J − I).

Proof : Since (X, B) is a symmetric design, any two blocks have λ common points

in any two blocks are adjacent.

Theorem 3.2. Let (X, B) be a quasi-symmetric 2 − (v, k, λ) design with sizes x and

y of intersections of blocks, then

1. the block graph is a SR graph (v0, R, λ0, µ) with (v0, R, λ0, µ) = (b, R, (θ

1+ θ2) + f (R) b , f (R) b ), where R = k(r − 1) − x(b − 1) (y − x) = −θ1θ2+ f (R) b , θ1 = r − λ − k + x (y − x) , θ2 = x − k

y − x, are three distinct eigenvalues of A, f (R) = (R − θ1)(R − θ2), b =

λv(v − 1) k(k − 1) , and

2. its block multigraph is a SR multigraph (m, n, µ, γ, R) with (m, n, µ, γ, R) = (k, r − λ, λk2, k(k − 1)(λ − 1), k(r − 1)).

Proof : To prove 1, let N be the v × b ncidence matrix of the design and A be

the adjacency matrix of its block graph Γ. We have (using the parameters v, k, b, r,

λ of the 2 - design):

NNT = (r − λ)I + λJ, NTN = kI + yA + x(J − I − A).

We know that both NNT and NTN have all-one eigenvectors j with eigenvalue kr.

Also, we know that NNT has only the eigenvalue r − λ on jT, with multiplicity

v − 1. Therefore NTN has this same eigenvalue, with the same multiplicity, and the

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and NTN. Therefore A has eigenvector j and only two eigenvalues on the space jT.

They are (r − λ − k + x)/(y − x) with multiplicity v − 1 and (x − k)/(y − x) with multiplicity b − v. By our observation above, Γ is an SR graph.

A = 1 (y − x)N TN −(k − x) (y − x)I − x (y − x)J (∗) Multiplying both sides of (*) by J, and J2 = bJ, then

AJ = RJ = kr (y − x)J − (k − x) (y − x)J − xb (y − x)J ⇒ R = kr − (k − x) − xb (y − x) = k(r − 1) − x(b − 1) (y − x) Since R, θ1 = r − λ − k + x (y − x) , θ2 = x − k

(y − x) are three distinct eigenvalues of A Let f (x) = (x − θ1)(x − θ2) = x2− (θ1+ θ2)x + θ1θ2 Then A2 = (θ 1+ θ2)A − θ1θ2I + f (R) b J, where b = λv(v − 1) k(k − 1) Thus λ0 = (θ 1+ θ2) + f (R) b , µ = f (R) b .

To prove 2, let M = NTN − kI = yA + x(J − I − A) be a adjacency matrix of a

block multigraph of the design

MJ = (NTN − kI)J = k(r − 1)J = RJ ⇒ R = k(r − 1) M2 = (NTN − kI)2 = NTNNTN − 2kNTN + k2I = (r − λ − 2k)(NTN − kI) + k(r − λ − k)I + λk2J = (r − λ − 2k)M + k(r − λ − k)I + λk2J

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m = k, n = r − λ, µ = λk2, γ = m(n − m) + µ − R = k(k − 1)(λ − 1).

Theorem 3.3. [7] The block multigraph of a 2-(v, k, λ) design is an SR multigraph with (m, n, µ, γ, R) = (k, r − λ, λk2, k(k − 1)(λ − 1), k(r − 1)).

Proof : Similarly to Theorem 3.2 (2).

Remark: when are the above multigraphs simple graphs? Are they designs with some interests?

Seven examples of 2-designs together with the related graphs and multigraphs are given below: 2-(v, k, λ) SR(m, n, µ, γ, R) 2 - (9, 3, 1) SR(3, 3, 9, 0, 9) 2 - (6, 3, 2) SR(3, 3, 18, 6, 12) 2 - (8, 4, 3) SR(4, 4, 48, 24, 24) 2 - (10, 4, 2) SR(4, 4, 32, 12, 20) 2 - (16, 4, 1) SR(4, 4, 16, 0, 16) 2 - (16, 6, 2) SR(6, 4, 72, 30, 30) 2 - (16, 6, 3) SR(6, 6, 108, 60, 48) Example 1: 2 - (6, 3, 2) design

Let X = {1, 2, 3, 4, 5, 6}, then (X, B) is a 2 - (6, 3, 2) design where B = {Bx|1 ≤ x ≤

10},

B1 = {1, 2, 3}, B2 = {1, 2, 4}, B3 = {1, 3, 5}, B4 = {1, 4, 6}, B5 = {1, 5, 6},

B6 = {2, 3, 6}, B7 = {2, 4, 5}, B8 = {2, 5, 6}, B9 = {3, 4, 5}, B10= {3, 4, 6}.

Note that |Bi∩ Bj| = 1 or 2 for distinct 1 ≤ i, j ≤ 10, that is it is quasi-symmetric.

Note also that r = λ(v − 1) (k − 1) =

2 · (6 − 1) (3 − 1) = 5.

The block multigraph of Γ the above 2 - (6, 3, 2) design of order n = r − λ = 3 is an SR multigraph (m, n, µ, γ, R) = (k, r − λ, λk2, k(k − 1)(λ − 1), k(r − 1)) =

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(3, 3, 18, 6, 12). The block graph of the 2 - (6, 3, 2) design is the Petersen graph(see Figure 1).

Example 2: 2 - (8, 4, 3) design

Let X = {0, 1, 2, 3, 4, 5, 6, 7}, then (X, B) is a 2 - (8, 4, 3) design where B = {Bx|1 ≤

x ≤ 14},

B1 = {0, 1, 2, 3}, B2 = {0, 1, 2, 4}, B3 = {0, 1, 5, 6}, B4 = {0, 2, 5, 7}, B5 = {0, 3, 4, 5},

B6 = {0, 3, 6, 7}, B7 = {0, 4, 6, 7}, B8 = {1, 2, 6, 7}, B9 = {1, 3, 4, 6}, B10 = {1, 3, 5, 7},

B11= {1, 4, 5, 7}, B12 = {2, 3, 4, 7}, B13 = {2, 3, 5, 6}, B14= {2, 4, 5, 6}.

Note that r = 3 · (8 − 1)

(4 − 1) = 7. The block multigraph Γ of 2 - (8, 4, 3) design of order

n = 4 is an SR multigraph (4, 4, 48, 24, 24).

Example 3: 2 - (9, 3, 1) design

Let X = {1, 2, 3, . . . , 9}, and those 9 elements are arranged in a 3 × 3 array as show below:

1 2 3 4 5 6 7 8 9

Then (X, B) is a 2 - (9, 3, 1) design where B = {Bx|1 ≤ x ≤ 12}, and

slope 0, B1 = {1, 2, 3}, B2 = {4, 5, 6}, B3 = {7, 8, 9},

slope ∞, B4 = {1, 4, 7}, B5 = {2, 5, 8}, B6 = {3, 6, 9},

slope 1, B7 = {1, 6, 8}, B8 = {2, 4, 9}, B9 = {3, 5, 7},

slope −1, B10= {1, 5, 9}, B11= {2, 6, 7}, B12= {3, 4, 8}.

The block multigraph Γ of the above 2 - (9, 3, 1) design of order n = 3 is an SR multigraph (3, 3, 9, 0, 9).

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(v, k, λ, µ) = (R(R − n + 2m − µ − 1)

µ + R + 1, R, n − 2m + µ, µ) = (12, 9, 6, 9).

Example 4: a 2 - (10, 4, 2) design

Let X = {0, 1, 2, . . . , 9} and B = {Bx|1 ≤ x ≤ 15}, where

B1 = {0, 1, 2, 3}, B2 = {0, 1, 4, 5}, B3 = {0, 2, 4, 6}, B4 = {0, 3, 7, 8},

B5 = {0, 5, 7, 9}, B6 = {0, 6, 8, 9}, B7 = {1, 2, 7, 8}, B8 = {1, 3, 6, 9},

B9 = {1, 4, 7, 9}, B10= {1, 5, 6, 8}, B11= {2, 3, 5, 9}, B12= {2, 4, 8, 9},

B13= {2, 5, 6, 7}, B14 = {3, 4, 5, 8}, B15 = {3, 4, 6, 7}.

Note that |Bi ∩ Bj| = 1 or 2 for distinct 1 ≤ i, j ≤ 15, and hence it is a

quasi-symmetric design; note also that r = 6. The block multigraph Γ of 2 - (10, 4, 2) design of order n = 4 is an SR multigraph (4, 4, 32, 12, 20).

Example 5: a 2 - (16, 4, 1) design

Let X = {0, 1, 2, . . . , 9, a, b, . . . , f }, then (X, B) is a 2 - (16, 4, 1) design where

B = {Bx|1 ≤ x ≤ 20}, B1 = {0, 1, 2, 3}, B2 = {0, 4, 5, 6}, B3 = {0, 7, 8, 9}, B4 = {0, a, b, c}, B5 = {0, d, e, f }, B6 = {1, 4, 7, a}, B7 = {1, 5, b, d}, B8 = {1, 6, 8, e}, B9 = {1, 9, c, f }, B10= {2, 4, c, e}, B11= {2, 5, 7, f }, B12 = {2, 6, 9, b}, B13= {2, 8, a, d}, B14 = {3, 4, 9, d}, B15 = {3, 5, 8, c}, B16= {3, 6, a, f }, B17= {3, 7, b, e}, B18 = {4, 8, b, f }, B19= {5, 9, a, e}, B20 = {6, 7, c, d},

Note that |Bi ∩ Bj| = 0 or 1 for distinct 1 ≤ i, j ≤ 20, and hence it is a

quasi-symmetric design. r = 5. The block multigraph Γ of the above 2 - (16, 4, 1) design of order n = 4 is an SR multigraph (4, 4, 16, 0, 16).

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Example 6: a 2-(16, 6, 2) design

Let X = {0, 1, 2, . . . , 15}, and those 16 elements are arranged in a 4 × 4 array A as shown below:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

For each x, 0 ≤ x ≤ 15, we define a block Bx consisting of the elements in the same

row or column of A as x, excluding x. Then (X, B) is a 2-(16, 6, 2) design where

B = {Bx|0 ≤ x ≤ 15}. More precisely, 2 - (16, 6, 2) design

B0 = {1, 2, 3, 4, 8, 12}, B1 = {0, 2, 3, 5, 9, 13}, B2 = {0, 1, 3, 6, 10, 14}, B3 = {0, 1, 2, 7, 11, 15},

B4 = {5, 6, 7, 0, 8, 12}, B5 = {4, 6, 7, 1, 9, 13}, B6 = {4, 5, 7, 2, 10, 14}, B7 = {4, 5, 6, 3, 11, 15},

B8 = {9, 10, 11, 0, 4, 12}, B9 = {8, 10, 11, 1, 5, 13}, B10= {8, 9, 11, 2, 6, 14},

B11= {8, 9, 10, 3, 7, 15}, B12 = {13, 14, 15, 0, 4, 8}, B13= {12, 14, 15, 1, 5, 9},

B14= {12, 13, 15, 2, 6, 10}, B15= {12, 13, 14, 3, 7, 11}.

The block multigraph Γ of the above 2 - (16, 6, 2) design of order n = 4 is an SR multigraph (6, 4, 72, 30, 30). Moreover, since it is symmetric, the adjacency matrix

A of Γ is A =    0 2 . .. 2 0    16×16 = 2J − 2I Example 7: a 2-(16, 6, 3) design

Let X = {1, 2, . . . , 24}, then (X, B) is a 2 - (16, 6, 3) design with B = {Bx|0 ≤ x ≤

24}, where

(i) B1 = {1, 2, 5, 6, 9, 10}, B2 = {1, 3, 5, 7, 9, 11}, B3 = {1, 4, 5, 8, 9, 12};

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(ii) B7 = {1, 2, 7, 8, 15, 16}, B8 = {1, 3, 6, 8, 14, 16}, B9 = {1, 4, 6, 7, 14, 15}; B10= {3, 4, 5, 6, 13, 14}, B11= {2, 4, 5, 7, 13, 15}, B12 = {2, 3, 5, 8, 13, 16}; (iii) B13= {1, 2, 11, 12, 13, 14}, B14= {1, 3, 10, 12, 13, 15}, B15 = {1, 4, 10, 11, 13, 16}; B16= {3, 4, 9, 10, 15, 16}, B17 = {2, 4, 9, 11, 14, 16}, B18= {2, 3, 9, 12, 14, 15}; (iv) B19 = {5, 6, 11, 12, 15, 16}, B20 = {5, 7, 10, 12, 14, 16}, B21= {5, 8, 10, 11, 14, 15}; B22= {7, 8, 9, 10, 13, 14}, B23 = {6, 8, 9, 11, 13, 15}, B24= {6, 7, 9, 12, 13, 16}.

The 6 blocks {B1, B2, B3, B4, B5, B6} in case (i) satisfying the conditions that

1. |Bi ∩ Bj| =

½

0 if |i − j| = 3

3 if |i − j| 6= 3 for i 6= j, i, j ∈ {1, 2, . . . , 6}

2. Each block Bi of (i) meet 2 points |Bi∩ Bj| = 2 for Bk is any block of (ii), (iii),

or (iv).

Similarly conclusion hold for cases (ii), (iii), and (iv).

Then very vertex (i.e., a block) lies on 3 × 4 + 2 × 18 = 48 edges, the block multigraph of the above 2 - (16, 6, 3) design of order is an SR multigraph (6, 6, 108, 60, 48).

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4

Regular graphs of 3 or 4 distinct eigenvalues

Lemma 4.1. Let Γ be a graph which is not complete or empty, with adjacency matrix

A Then Γ is an SR graph if and only if A2 is a linear combination of A, I and J.

Proof : The ij - entry of A2 is equal to the number of walks of length two from i

to j in Γ. If Γ is an SR graph with parameters k, λ, µ according as i and j are equal, adjacent or distinct and non - adjacent, hence A2 = kI + λI + µ(J − I − A).

Conversely, if A2 is a linear combination of A, I and J. A2 = m

1A + m2I + m3A this

number are (m2 + m3), (m1 + m3), m3 according as i and j are equal, adjacent or

distinct and non-adjacent. Hence Γ is an SR graph.

Lemma 4.2. If Γ is a connected graph with diameter d then A(Γ) has at least d+1 distinct eigenvalues, or equivalently if Γ is a graph with d+1 distinct eigenvalues, the the diameter of Γ is at most d.

Proof : Suppose A=A(Γ) has distinct eigenvalues θ0, θ1, θ2, · · · , θm where m < d.

Then m(x) =

m

Y

i=0

(x−θi) is the minimal polynomial of A, and hence Ad−(m+1)·m(A) = 0,

then we have Ad = C

d−1Ad−1+ Cd−2Ad−2+ · · · + C1A + C0I for some Ci ∈ R. For

two vertices x, y ∈ V (Γ) with ∂(x, y) = d, the xy position in the above equation and 0 6= (Ad)

xy = Cd−1(Ad−1)xy+Cd−2(Ad−2)xy+· · ·+C1Axy+C0Ixy = 0, a contradiction;

so A(Γ) has at least d + 1 distinct eigenvalues.

Theorem 4.1. Let Γ be a connected k -regular graph with s distinct eigenvalues,

1. if s = 2, then Γ is complete graph.

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Proof : To prove 1, by above lemma, the diameter d(Γ) of Γ is 0 or 1. Since Γ is

connected, d(Γ) 6= 0, hence d(Γ) = 1, and Γ is a complete graph. To prove 2, let A be an adjacency matrix of Γ, and

f (x) = (x − θ1)(x − θ2) = x2− (θ1+ θ2)x + θ1θ2,

Then A2 = (θ

1+ θ2)A − θ1θ2I +

f (k)

n J. Hence Γ is an SR graph with (v, k, λ, µ) =

(v, −θ1θ2 +

f (k)

v , θ1+ θ2+ f (k)

v , µ).

Theorem 4.2. Let Γ be a connected k -regular graph with 4 distinct eigenvalues

k > θ1 > θ2 > θ3, then

1. Γ is walk regular;

2. the diameter of Γ is 3 if the number of vertices of the graph is more than

α +pα2− 4β 2 or less than α −pα2 − 4β 2 , where α = 1 + k2− (θ 1+ θ2+ θ3)k − θ1θ2θ3,and β = (k − θ1)(k − θ2)(k − θ3).

Proof : To prove 1, let Γ be a regular graph with 4 distinct eigenvalues, and A

be an adjacency matrix of Γ. f (x) = (x − θ1)(x − θ2)(x − θ3) = x3− (θ1+ θ2+ θ3)x2+ (θ1θ2+ θ2θ3+ θ3θ1)x − θ1θ2θ3 Let α0 = θ123, α1 = θ1θ22θ33θ1, α2 = θ1θ2θ3 and β = (k−θ1)(k−θ2)(k−θ3), then A3 − α 0A2+ α1A − α2I = β nJ (*), where n = |V (Γ)|, and hence A3 = α 0A2− α1A + α2I +β nJ.

Since the diagonal entries of A2, A, I, J are constant, the diagonal entries of A3 are

constant α0k +α2+

β

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for the diagonal entries of A4, A5, A6, · · · , etc are constant. Hence all regular graphs

with 4 distinct eigenvalues are walk regular.

To prove 2, let A be an adjacency matrix of Γ, and

f (x) = (x − θ1)(x − θ2)(x − θ3) = x3− (θ 1+ θ2+ θ3)x2 + (θ1θ2+ θ2θ3+ θ3θ1) − θ1θ2θ3 Then A3− (θ 1+ θ2+ θ3)A2+ (θ1θ2+ θ2θ3+ θ3θ1)A − θ1θ2θ3I = β nJ(*)

and hence the diameter of Γ is at most 3.

We will claim that k3(x) > 0 for each x ∈ V (Γ) under the numerical constraints.

For x ∈ V (Γ), we first evaluate A3

xx: A3 xx = X y∈Γ1(x) A2 xx by definition, and A3 xx = f (k) n + (θ1+ θ2+ θ3)A 2 xx− (θ1θ2+ θ2θ3+ θ3θ1)Axx+ θ1θ2θ3Ixx = f (k) n + (θ1+ θ2+ θ3)k + θ1θ2θ3 by (*). Hence, X y∈V (Γ) A2 xy = (A2J)xx = k2, X y∈Γ1(x) Axy = A2xx = k, X y∈Γ1(x) A2xy = A3xx = β n + (θ1+ θ2+ θ3)k + θ1θ2θ3,

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and then X y∈Γ2(x) A2xy = X y∈V (Γ) A2xy − A2xx X y∈Γ1(x) A2xy X y∈Γ3(x) A2xy = k2− k −β n − (θ1+ θ2 + θ3)k − θ1θ2θ3 ≥ k2(x). It follows that 1 + k + k2(x) ≤ 1 + k2 − k − β n − (θ1+ θ2 + θ3)k − θ1θ2θ3 = α − β n, whenever n > α + p α2− 4β 2 or n < α −pα2 − 4β 2 , as required.

Theorem 4.3. [2] Let Γ be a connected k -regular graph on v vertices with Spec(Γ)=(k1, θm1

1 , θm22, θ3m3). Then

1. m1 = m2 = m3 = (v − 1)/3 and k = (v − 1)/3 or 2(v − 1)/3, or

2. Γ has two or four integral eigenvalues.

Moreover, if Γ has exactly two integral eigenvalues, then the other two have the same mulitplicities and are of the form 1

2(a ±

b) for a, b ∈ Z.

Theorem 4.4. Let Γ be a connected regular graph on v vertices with four distinct eigenvalues, say Spec(Γ) = (k1, θm1

1 , θ2m2, θ3m3).

Let λ = (k3 + m

1θ13 + m2θ23) + m3θ33/vk. Then Γ is distance-regular if and only if

for every vertex x the number of vertices k2(x) at distance two from x is

k2(x) =

k(k − 1 − λ)2

(k − λ)(λ − k − (θ1+ θ2+ θ3) − (θ1θ2+ θ2θ3+ θ3θ1) + θ1θ2θ3)

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It was conjectured by van Dam [2] that the proposition was also true without the conditions for k2(x) , i.e., that for every connected regular graph with four distinct

eigenvalues we have that the number of vertices k2 at distance two from a given vertex

is at least k2(x).

The following are examples of some connected regular graphs with 4 distinct eigenvalues and with diameter 2, all of them are walk-regular, though some of them are not distance regular.

Example 1: Γ = C6, the complement of C6(see Figure 2)

A(C6) =         0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0        

the characteristic polynomial of A(C6) is f (x) = x2(x − 1)(x − 3)(x + 2)2, and

Spec(C6) = (31, 11, 02, −22), this graph is walk regular.

Example 2: Γ = Q3, the complement of Q3(see Figure 3)

A(Q3) =             0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0            

the characteristic polynomial of A(Q3) is f (x) = x3(x − 2)(x − 4)(x + 2)3, and

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Example 3: Γ = 2C4, the complement of 2C4(see Figure 4) A(2C4) =             0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 0 0            

the characteristic polynomial of A(2C4) is f (x) = (x − 5)(x − 1)2(x + 1)4(x + 3), and

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References

[1] R. C. Bose, S. S. Shrikhande and N. M. Singhi, Edge regular multigraphs and partial geometric designs with an application to the embedding of quasi-residual designs, Coll. Int. Sul. Teo. Combin, Tom I (Acc Naz. Lin-cei, Roma, 1976) 49-81.

[2] E.R. van Dam, Regular graphs with four eigenvalues, Linear Alg. Appl. 226-228 (1995) 139-162.

[3] E.R. van Dam and E. Spence, Combianatorial designs with two singular values. I. Uniform multiplicative designs, J. Combin. Theory A 107 (2004) 127-142.

[4] E.R. van Dam and E. Spence, Combinatorial designs with two singular values II Partial geometric designs, Linear Algebra and its Applications 396 (2005) 303-316.

[5] Tayuan Huang, A technique of Spectral Characterizations, preprint 1995.

[6] K. Metsch, Quasi-residual designs, 1 -designs, and strongly regular multi-graphs, Discrete Mathematics 143 (1995) 167-188.

[7] A. Neumaier, Quasi-residual designs, 1 -designs, and strongly regular multigraphs,Geom. Dedicata 12 (1982) 351-366.

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B

3

B

1

B

2

B

7

B

9

B

1 0

B

8

B

4

B

6

B

5 Figure 1 1 2 3 4 5 6 Figure 2

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1 2 3 4 5 6 7 8 Figure 3 1 2 3 4 5 7 6 8 Figure 4

Figure

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References