國
立
交
通
大
學
應用數學系
碩
士
論
文
由局限距離正則圖構造強正則圖的一種方法
Construct Strongly Regular Graphs from
D-bounded Distance-Regular Graphs
研 究 生:曾雅榕
由局限距離正則圖構造強正則圖的一種方法
Construct Strongly Regular Graphs from
D-bounded Distance-Regular Graphs
研 究 生 : 曾雅榕 Student : Ya-Jung Tseng
指導教授 : 翁志文 Advisor : Chih-Wen Weng
國 立 交 通 大 學
應 用 數 學 系
碩 士 論 文
A Thesis
Submitted to Department of Applied Mathematics
College of Science National Chiao Tung University
In partial Fulfillment of Requirement
For the Degree of Master In Applied Mathematics
June 2008
Hsinchu, Taiwan, Republic of China
由局限距離正則圖構造強正則圖的一種方法
研究生 : 曾雅榕 指導教授 : 翁志文
國 立 交 通 大 學
應 用 數 學 系
摘要
在數學的領域中,透過不同數學方法而得到相同的理論是常見
的。本論文的目的是探討,透過不同的兩個方法來証明強正則圖的參
數, 一個利用變數的計算(如定理 7.6 所示),另一個是利用線性代數
的方法(如定理 8.3). 結果顯示此強正則圖的參數呈現是一樣的。
Construct Strongly Regular Graphs from D-bounded
Distance-Regular Graphs
Student : Ya- Jung Tseng Advisor : Chih-Wen Weng
Department of Applied Mathematics
National Chiao Tung University
Abstract
In the field of mathematics, it is common to achieve the same
conclusion of the theory via various approaches. The purpose of this
thesis is to probe the parameters of a strongly regular graph via two
different methods, one (Theorem 7.6) with the use of counting
argument and the other (Theorem 8.3) with a linear algebric method.
The result shows that these parameters are determined explicitly and
are the same.
Contents
Abstract ( in Chinese ) i
Abstract ( in English )
ii
Contents iii
1 Introduction 1
2 Preliminaries of graphs 3
3 Distance-regular graphs
5
4 Preliminaries of designs 9
5 Quasisymmetric designs and strongly regular graphs 11
6 D-bounded distance-regular graphs 17
7 Construct strongly regular graphs from D-bounded
19
distance-regular graphs
8 Another proof of Theorem 7.6 25
Reference 30
1
Introduction
In mathematics, it is common to get the same theory through different approaches. In this thesis, we adopt two different methods to construct a strongly regular graph, one (Theorem 7.6) with the use of counting argument and the other (Theorem 8.3) with a linear algebric method.
We consider a regular graph Γ, called a D-bounded distance-regular graph. This graph Γ contains many weak-geodetically closed sub-graphs. We fix a weak-geodetically closed subgraph ∆ of diameter s and let P (resp. B) be the set of all weak-geodetically closed subgraphs containing ∆ of diameter s + 1 (resp. s + 2). We show that (P, B) is a 2-design. We show that the block graph of (P, B) is a strongly regular graph and determine its parameters explicitly by two methods.
To do this, we need some concepts about graphs and designs. Hence, in Chapter 2, we review some definitions and basic concepts of graphs, such as regular graphs, subgraphs, eigenvalues, etc. In Example 2.2, a special graph, Petersen graph, is introduced since it is an essential example in graph theory.
Next, Chapter 3 introduces the distance-regular graphs along with a spe-cial class of distance-regular graphs, the class of strongly regular graphs, which will be used in Chapter 5. Meanwhile, a classic theorem and its proof, Theorem 3.5, are mentioned to characterize strongly regular graph by its eigenvalues.
In Chapter 4, we give some definitions of designs and its basic concepts. Firstly, a t-design is presented accompanied with a special case, 2-design later utilized in the following chapter.
Chapter 5 is about Quasisymmetric Designs and its relation to strongly regular graphs. Quasisymmetric Design is known as a 2-design (P, B), which constructs a graph with vertex set B. In Theorem 5.4, we know that this graph is a strongly regular graph. Next, in Lemma 5.5, its parameters are determined explicitly. Within Lemma 5.5, a special case, Corollay 5.6, will be generated as well and used in Theorem 8.3.
In Chapter 6, we give some definitions and properties about D-bounded distance-regular, which will be used in the next Chapter.
In Chapter 7, a strongly regular graph is constructed from a D-bounded distance-regular graph. Firstly, we define a graph G(∆, 2) which is known to be either a clique or a strongly regular graph in Theorem 7.1. The parame-ters of a strongly regular graph will be obtained. It is worth noticing that Theorem 7.6 is the main theorem in this thesis.
Finally, in Chaptet 8, we use a linear algebraic method to prove Theorem 7.6 again as Theorem 8.3 shown.
2
Preliminaries of graphs
In this section we review some definitions and basic concepts of graphs. The reader can refer to [13] for more details.
Definition 2.1. A graph G is a pair consisting of a vertex set V (G) and an edge set E(G), where E(G) is a set containing some 2-subsets of V (G). For a vertex u ∈ V (G) and an edge uv ∈ E(G), we say u is incident to uv and u is adjacent to v.
Example 2.2. Let V (G) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and E(G) = {01, 12, 23, 34, 40, 05, 16, 27, 38, 49, 57, 58, 69, 68, 79}. Then G = (V (G), E(G)) is a graph. See Figure 1 for the drawing of this graph. G is called the Petersen graph.
s s s s s s s s s s @ @ @ @ @ @ @ @ @ H H H H H H A A A A A A 0 1 2 3 4 5 6 7 8 9
Figure 1: Petersen graph
Definition 2.3. The degree of vertex v in a graph G is the number of edges incident to v. G is regular if every vertex has the same degree. It is b0-regular
if the common degree is b0.
Definition 2.5. A path in a graph G is a sequence of vertices so that any two consecutive vertices are adjacent. For two vertices u, v in G, a u, v-path is a path with the first vertex u and the last vertex v.
Definition 2.6. A graph G is connected if each pair of vertices in G belongs to a path.
Definition 2.7. If G has a u, v-path, then the distance from u to v, written ∂G(u, v) or simply ∂(u, v), is the least length of a u, v-path. If G has no such
path, then ∂(u, v) = ∞. The diameter (diam G) is max{∂(u, v)|u, v ∈ V (G)}.
Example 2.8. The Petersen graph has diameter 2, since nonadjacent ver-tices have a common neighbor.
Definition 2.9. A subgraph of a graph G is a graph H such that V (H) ⊆ V (G) and E(H) ⊆ E(G). An induced subgraph is a subgraph H of a graph G such that if u, v ∈ V (H) and uv ∈ E(G) then uv ∈ E(H).
Definition 2.10. Let G be a graph with vertex set V (G) of size n and edge set E(G) of size m. The adjacency matrix of G, written A(G), is the n-by-n matrix in which entry aij, where i, j ∈ V (G), is 1 if ij ∈ E(G) and otherwise
is 0. The incidence matrix M (G) is the n-by-m matrix in which entry mie,
where i ∈ V (G) and e ∈ E(G), is 1 if i is incident to e and otherwise is 0. Definition 2.11. The eigenvalues of a graph G are the eigenvalues of its adjacency matrix A(G).
Definition 2.12. The complement G of a graph G is the graph with vertex set V (G)=V (G) and the edge set E(G) = {uv | u, v ∈ V (G), uv /∈ E(G)}.
Definition 2.13. Assume G is a connected graph with diameter D. For all vertices x in G and for 0 ≤ i ≤ D, set
Gi(x) := {y ∈ G | ∂(x, y) = i}.
3
Distance-regular graphs
Assume Γ is a connected graph with diameter D. Γ is said to be distance-regular whenever for 0 ≤ h, i, j ≤ D and for vertices x, y in Γ with ∂(x, y) = h, the number
phij :=Γi(x) ∩ Γj(y)
is independent of x, y. The constants ph
ij are known as the intersection
num-bers of Γ. For convenience, set ci := pi1i−1, ai := pi1i, bi := pi1i+1 and ki := p0ii.
Note that c1 = 1, a0 = 0, bD = 0 and
k1 = ci+ ai+ bi for 0 ≤ i ≤ D.
The Petersen graph described in Figure 1 is a distance-regular graph with diameter D = 2 and intersection numbers c1 = 1, c2 = 1, a1 = 0, a2 = 2, b0 =
3, b1 = 2.
Next, we give a special class of distance-regular graphs. Its diameter is two.
Definition 3.1. A strongly regular graph SRG(v, b0, a1, c2) is a b0-regular
graph which has v vertices and the following properties hold:
(i) For any two adjacent vertices x, y, there are exactly a1 vertices adjacent
to x and to y.
(ii) For any two nonadjacent vertices x, y, there are exactly c2 vertices
adjacent to x and y. Next, we give two examples.
Example 3.2. A pentagon is an SRG(5, 2, 0, 1).
Example 3.3. The Petersen graph is an SRG(10, 3, 0, 1).
Next, we introduce the lemma, which will be used later.
Lemma 3.4. A connected graph with diameter d has at least d + 1 distinct eigenvalues.
Proof. Let A = A(G) be the adjacency matrix. Suppose A has distinct eigenvalues θ0, θ1, ..., θm where m < d. Then m(x) =
Qm
i=0(x − θi) is the
minimal polynomial of A. Hence Ad−m−1m(A) = 0. Expanding to find
Ad= Cd−1Ad−1+ Cd−2Ad−2+ ... + C0 for some Ci ∈ R.
Pick two vertices x, y ∈ G with ∂(x, y) = d. We check the xy position in the above equation and find
0 6= (Ad)xy = (Cd−1Axyd−1+ Cd−2Ad−2xy + ... + C0) = 0,
The following theorem characterize a strongly regular graph by its eigen-values.
Theorem 3.5. ([1, Problem 31H]). Let G denote a connected b0-regular graph
of diameter 2. Then G is a strongly regular graph SRG(v, b0, a1, c2) for some
scalars v, b0, a1, c2 if and only if G has three distinct eigenvalues.
Proof. (⇒)Let A = A(G) be the adjacency matrix. Observe A 1 1 .. . 1 = b0 b0 .. . b0 = b0 1 1 .. . 1 . Claim(i): A2 + (c
2 − a1)A + (c2 − b0)I = c2J, where J denotes the all 1’s
matrix.
Pick x, y ∈ G. Compare x, y entries on both sides, according to the following three cases. Case 1. ∂(x, y) = 0 : b0+ 0 + (c2− b0) · 1 = c2; Case 2. ∂(x, y) = 1 : a1+ (c2− a1) · 1 + 0 = c2; Case 3. ∂(x, y) = 2 : c2+ 0 + 0 = c2.
The left hand side acts on (1, 1, . . . , 1)t is 0. Let u = (1, 1, . . . , 1)t. Other
eigenvectors are orthogonal to u. Hence by claim(i)
(A − b0I)(A2+ (c2− a1)A + (c2− b0)I)u = (A − b0I)(c2J )u
= 0.
By claim(ii), A has eigenvalues among b0, (a1−c2±p(c2− a1)2− 4(c2− b0))/2.
By Lemma 3.4, G has at least three eigenvalues. Hence G has exactly three eigenvalues.
(⇐) Clearly, G has a eigenvalue b0. Suppose g < s are the other two.
Then (A − gI)(A − sI) = (b0− g)(b0− s)J/|G|. (As above, apply both sides
to all eigenvectors of A.) Pick x, y ∈ G. Case 1. ∂(x, y) = 0 : A2xy = (g + s)Axy− gsIxy + (b0 − g)(b0− s) |G| Jxy = −gs +(b0− g)(b0− s) |G| is independent of x = y, and b0 = −gs + (b0 − g)(b0− s)/|G|. Case 2. ∂(x, y) = 1 : A2xy = (g + s)Axy− gsIxy + (b0 − g)(b0− s) |G| Jxy = g + s + (b0 − g)(b0− s) |G|
is independent of x, y with ∂(x, y) = 1 and a1 = g + s + (b0− g)(b0− s)/|G|.
Case 3. ∂(x, y) = 2 : A2xy = (g + s)Axy− gsIxy + (b0 − g)(b0− s) |G| Jxy = (b0− g)(b0− s) |G|
is independent of x, y with ∂(x, y) = 2 and c2 = (b0− g)(b0 − s)/|G|.
G has diameter 2. Since G has three eigenvalues. Hence b1 := b0− a1− 1
is |G1(x) ∩ G2(y)| for any x, y ∈ G with ∂(x, y) = 1. Then G is a strongly
regular graph.
Example 3.6. The Petersen graph is a 3-regular graph of diameter 2. And from Example 3.3, we know it is a strongly regular graph . By Theorem 3.5, the Petersen graph has three distinct eigenvalues.
4
Preliminaries of designs
We first give the definition of a design and its basic concepts.
Definition 4.1. (P, B) is a t-(v, k, λ) design whenever the following (i)-(iv) hold.
(i) P is a finite set of v elements.
(ii) B is a class of subsets, called blocks, of P. (iii) |B| = k for all B ∈ B.
(iv) For any t distinct elements p1, p2, ..., pt ∈ P there are exactly λ blocks
B1, B2, ..., Bλ ∈ B such that pi ∈ Bj for all i, j.
A t-design is a t-(v, k, λ) design for some positive integers v,k,λ.
Example 4.2. P = {1, 2, 3, . . . , n}, B = {P }. Then (P, B) is a t-(n, n, 1) design for any t = 1, 2, . . . , n.
Example 4.3. P = {0, 1}2 = {0, 1}×{0, 1}. B = {{(0, 0), (1, 0)}, {(0, 1), (1, 1)},
{(0, 0), (0, 1)}, {(1, 0), (1, 1)}, {(0, 0), (1, 1)}, {(1, 0), (0, 1)}}. Then (P, B) is a 2-(4, 2, 1) design.
We give a few properties of 2-design which will be used later. Lemma 4.4. ([1, Theorem 19.2]). In a 2-(v, k, λ) design, there are b := λ v 2 / k 2 blocks.
Proof. We count the pairs (S, B), where S ⊆ B ∈ B and |S| = 2, by two ways: v 2 λ = b k 2 . Hence b = λ v 2 / k 2 .
Lemma 4.5. ([1, Theorem 19.3]). For i = 0, 1, 2, any i points in a 2-(v, k, λ) design (P, B) are contained in γi := λ
v − i 2 − i / k − i 2 − i blocks.
Proof. Let I ⊆ P with |I| = i. Count the pair (S, B), where S ∪ I ⊆ B ∈ B, S ∩ I = φ and |S| = 2 − i, by two ways:
v − i 2 − i λ = γi k − i 2 − i . Hence γi = λ v − i 2 − i / k − i 2 − i .
Definition 4.6. In a 2-(v, k, λ) design, γ1 = λ(v − 1)/(k − 1) is usually
denoted by r, referred as the replication number.
We give the following example of 2-design. To calculate b and r.
Example 4.7. Let P = {1, 2, 3, 4, 5, 6, 7}, B = {{1, 2, 3}, {1, 6, 7}, {1, 4, 5}, {3, 4, 7}, {2, 5, 7}, {2, 4, 6}, {3, 5, 6}}. Then (P, B) is 2-(7, 3, 1) design. And b = 1 7 2 / 3 2 = 7, r = 1(7 − 1)/(3 − 1) = 3.
5
Quasisymmetric designs and strongly
reg-ular graphs
In this section, we introduce the concept of Quasisymmetric Designs and the graph which is constructed from a quasisymmetric design. This graph is known to be a strongly regular graph. We determine its parameters explicitly.
We give a special class of 2-design.
Definition 5.1. A quasisymmetric design (QSD) with parameters ρ < α is a 2-design (P, B) such that |B ∩ B0| = ρ or α for all distinct blocks B and B0
in B.
We construct a graph from a quasisymmetric design.
Definition 5.2. Suppose that (P, B) is a QSD with parameters ρ < α. Let G be a graph with vertex set B and two vertices B and B0 of B are adjacent if and only if |B ∩ B0| = ρ. Then G is called the block graph of (P ,B).
Next, we introduce the lemma, which will be used later.
Lemma 5.3. Let N denote the v ×b incidence matrix of (P, B) where v = |P | and b = |B| and J denote the all 1’s matrix, r is the replication number. Then N Nt= (r − λ)I + λJ. Proof. (N Nt)xy = X `∈B Nx`N`yt = X `∈B Nx`Ny` = r, if x = y ; λ, if x 6= y ; = ((r − λ)I + λJ )xy.
The following theorem is the main theorem in this section.
Theorem 5.4. ([1, Theorem 21.2]). The block graph G of a QSD (P, B) with parameters ρ < α is a strongly regular graph.
Proof. Let A(G) denote the adjacency matrix of G, and N denote the v × b incidence matrix of (P, B) where v = |P | and b = |B|. Let k be the cardinality of a block in B and J denotes the all 1’s matrix.
Claim: NtN = kI + ρA + α(J − I − A).
Case 1. B = B0 : k = |B ∩ B0| = k + 0 + α(1 − 1 − 0); Case 2. |B ∩ B0| = ρ : ρ = 0 + ρ + α(1 − 0 − 1); Case 3. |B ∩ B0| = α : α = 0 + 0 + α(1 − 0 − 0). The claim follows from this.
Note that A = (NtN + (α − k)I − αJ )/(ρ − α).
Recall that N Nt= (r − λ)I + λJ. We know that both N Ntand NtN have
all-one eigenvectors j(of different lengths!) with eigenvalue r − λ + λv (= kr). We know that N Nt has only the eigenvalue r − λ with multiplicity v − 1.
Therefore NtN has this same eigenvalue, with the same multiplicity, and the
eigenvalue 0 with multiplicity b − v. Observe
A 1 1 .. . 1 = kr + α − k − αb ρ − α 1 1 .. . 1 .
Hence G is regular. Observe the eigenvectors of NtN are eigenvectors of I
and J. Hence the other 2 eigenvalues of A are r − λ + α − k + 0
ρ − α ,
0 + α − k + 0 ρ − α .
Hence G has exactly three eigenvalues. By Theorem 3.5, we obtain G is a strongly regular graph.
In Theorem 5.4, we know ”G is a strongly regular regular.” Next, in the following lemma, we will determine its parameters explicitly.
Lemma 5.5. Let G denote the strongly regular graph obtained in Theorem 5.4 with parameters (v, b0, a1, c2). Then the following (i)-(iv) hold.
(i) v = |G|, (ii) b0 = kr + (α − k) − αb ρ − α , (iii) a1 = (ρ − α)(r − λ + 2(α − k) + kr + α − k − αb) + (α − k)(r − λ + α − k) (ρ − α)2 , (iv) c2 = (α − k)(r − λ + α − k) + (ρ − α)(kr + α − k − αb) (ρ − α)2 .
Where b = |B|, r is the replication number, and k is the cardinality of a block in B.
Proof. Clearly, G has a eigenvalue b0. Suppose g < s are the other two
eigenvalues of G. Where g = (r − λ + α − k)/(ρ − α), s = (α − k)/(ρ − α). (ii) Observe that
b0 =
kr + α − k − αb ρ − α
= −gs + (k − g)(k − s) | G | .
(iii) Observe that a1 = g + s + (k − g)(k − s) |G| = g + s + b0+ gs = r − λ + α − k ρ − α + α − k ρ − α + kr + α − k − αb ρ − α + (r − λ + α − k)(α − k) (ρ − α)2 = (ρ − α)(r − λ + 2(α − k) + kr + α − k − αb) + (α − k)(r − λ + α − k) (ρ − α)2 . (iv) Observe that
c2 = (k − g)(k − s) | G | = b0+ gs = kr + α − k − αb ρ − α + (r − λ + α − k)(α − k) (ρ − α)2 = (ρ − α)(kr + α − k − αb) + (α − k)(r − λ + α − k) (ρ − α)2 .
We give a special case of lemma 5.5, which will be used later. Corollary 5.6. From Lemma 5.5, and let ρ = 0, α = 1, λ = 1, we have
(ii) b0 = −kr + k − 1 + b;
(iii) a1 = 2k − 2kr − 2 + b + k2;
(iv) c2 = r − 2kr + k2+ b − 1.
(ii) We have b0 = kr + α − k − αb ρ − α = kr + 1 − k − b 0 − 1 = −kr − 1 + k + b. (iii) We have a1 = (ρ − α)(r − λ + 2(α − k) + kr + α − k − αb) + (α − k)(r − λ + α − k) (ρ − α)2 = (0 − 1)(r − 1 + 2(1 − k) + kr + 1 − k − b) + (1 − k)(r − 1 + 1 − k) (0 − 1)2 = (−1)(r − 1 + 2 − 2k + kr + 1 − k − b) + r − k − kr + k2 = (−1)(r + 2 − 3k + kr − b) + r − k − kr + k2 = −r − 2 + 3k − kr + b + r − k − kr + k2 = 2k − 2kr − 2 + b + k2. (iv) We have c2 = (α − k)(r − λ + α − k) + (ρ − α)(kr + α − k − αb) (ρ − α)2 = (1 − k)(r − 1 + 1 − k) + (0 − 1)(kr + 1 − k − b) (0 − 1)2 = (1 − k)(r − k) + (−1)(kr + 1 − k − b) = r − k − kr + k2 − kr − 1 + k + b = r − 2kr + k2+ b − 1.
6
D-bounded distance-regular graphs
Let Γ denote a distance-regular graph with diameter D ≥ 3. A sequence of vertices x, y, z of Γ is weak-geodetic whenever
∂(x, y) + ∂(y, z) ≤ ∂(x, z) + 1,
where ∂ is the distance function of Γ. A subgraph ∆ of Γ is weak-geodetically closed whenever for all weak-geodetic sequences of vertices x, y, z of Γ we have
x, z ∈ ∆ =⇒ y ∈ ∆.
Weak-geodetically closed subgraphs are called strongly closed subgraphs in [7]. We refer the reader to [9], [2], [5], [8], [10], [6] for the constructions of weak-geodetically closed subgraphs of Γ. It is immediate from the definition that a weak-geodetically closed subgraph ∆ is an induced subgraph of Γ and the distance function on ∆ is induced from that on Γ. Γ is D-bounded if (i) all of the weak-geodetically closed subgraphs of Γ are regular; and (ii) for all vertices x, y of Γ, x, y are contained in a common weak-geodetically closed subgraph ∆(x, y) of diameter ∂(x, y). In fact ∆(x, y) is uniquely determined by the vertices x and y [10, Corollary 5.4], and is distance-regular [10, Corol-lary 5.3]. Regular near polygons [2], [6], [9] and Hermitian forms graphs [5] are examples of bounded distance-regular graphs. The classification of D-bounded distance-regular graphs with some additional assumptions can be found in [11], [12].
Below we recall a few properties in a D-bounded distance-regular graph, which will be used in the next section. Let Γ denote a D-bounded distance-regular graph where D ≥ 3 is the diameter of Γ. Let ai, bi, ci denote the
intersection numbers of Γ for 0 ≤ i ≤ D. Let ∆ denote a weak-geodetically closed subgraph of diameter s for 0 ≤ s ≤ D. Note that ∆ is regular by the assumption (i) of D-bounded definition. In fact ∆ is distance-regular with intersection numbers
ai(∆) = ai(Γ)
ci(∆) = ci(Γ)
bi(∆) = bi(Γ) − bs(Γ)
for 0 ≤ i ≤ s [10, Corollary 5.3]. In particular a weak-geodetically closed subgraph of diameter 1 is a clique of size b0 − b1 + 1, and we refer such a
clique to a line. The intersection of weak-geodetically closed subgraphs is either an empty set or a weak-geodetically closed subgraph. Hence |∆ ∩ `| ∈ {0, 1, b0− b1+ 1} for any line ` in Γ. Let x denote a vertex in ∆. Then ∆1(x)
is a disjoint union of (b0− bs)/(b0 − b1) cliques of the form ` − {x}, where
` ⊆ ∆ is a line containing x. There are b0
b0− b1
− b0− bs b0− b1
lines `0 containing x such that `0 6⊆ ∆. For such a line `0, there exists a unique
weak-geodetically closed subgraph ∆0 of diameter s + 1 containing ∆ and `0. There are
b0 − bs+1
b0− b1
− b0− bs b0− b1
lines `00 (including `0) containing x such that `00− {x} ⊆ ∆0− ∆.
7
Construct strongly regular graphs from
D-bounded distance-regular graphs
Throughout the section, let Γ denote a D-bounded distance-regular graph with intersection numbers bi, ci for 0 ≤ i ≤ D. Note that bi > bi+1 for
0 ≤ i ≤ D − 1 [11, Lemma 2.6]. Fix an integer 0 ≤ s ≤ D − 3 and a weak-geodetically closed subgraph ∆ of Γ with diameter s. Let P = P(∆) denote the collection of weak-geodetically closed subgraphs containing ∆. If ∆ = {x} for some vertex x of Γ then we write P(x) for P(∆). It was shown that P is a ranked atomic lattice [3], where rank(Ω) is diameter(Ω) − s for Ω ∈ P. Let Pj = Pj(∆) denote the set of rank j elements in P for
0 ≤ j ≤ D − s. For each 1 ≤ i ≤ D − s we define a graph G(∆, i) whose vertex set is Pi, and vertex Ω is adjacent to vertex Ω0 in G(∆, i) if and only
if Ω ∩ Ω0 ∈ Pi−1, where Ω, Ω0 ∈ Pi.
Theorem 7.1. G(∆, 2) is either a clique or a strongly regular graph with parameters b0(G) = bs+2(bs− bs+2) (bs− bs+1)(bs+1− bs+2) , (7.1) a1(G) = ( bs+1− bs+2 bs− bs+1 )2+ bs+2 bs+1− bs+2 − 1, (7.2) c2(G) = ( bs− bs+2 bs− bs+1 )2. (7.3)
Proof. Fix x ∈ ∆ and Ω ∈ P2. Then x ∈ ∆ ⊆ Ω by the construction. First
we prove the number b0(G) = b0(G)(Ω) as expressed in (7.1). We do this
by counting the triples (Ω0, `, `0) in the order and its reversed order where Ω0 ∈ P2 such that Ω ∩ Ω0 ∈ P1, and `, `0 ⊆ Ω0 are lines containing x such that
` − {x} ⊆ Ω ∩ Ω0− ∆ and `0− {x} ⊆ Ω0 − Ω. We find b0(G) × ( b0− bs+1 b0− b1 − b0− bs b0 − b1 ) × (b0− bs+2 b0− b1 −b0− bs+1 b0− b1 ) = ( b0 b0− b1 − b0− bs+2 b0− b1 ) × (b0− bs+2 b0− b1 − b0− bs b0− b1 ) × 1 to obtain (7.1).
Second we fix another Ω0 ∈ P2 such that Ω ∩ Ω0 ∈ P1. We prove the
number a1(G) = a1(G)(Ω, Ω0) as expressed in (7.2). Let λ1 (resp. λ2) denote
the number of Ω00 ∈ P2 such that
Ω00∩ Ω = Ω00∩ Ω0 = Ω0∩ Ω (7.4) (resp.
Ω00∩ Ω ∈ P1, Ω00∩ Ω0 ∈ P1, Ω ∩ Ω0∩ Ω00 = ∆). (7.5)
Note that
a1(G) = λ1+ λ2. (7.6)
To determine λ1 we count the pairs (Ω00, `00) in the order and its reversed order,
where Ω00 ∈ P2 satisfies (7.4) and `00 ⊆ Ω00 is a line such that `00 6⊆ Ω ∪ Ω0. We
find λ1× ( b0− bs+2 b0− b1 − b0− bs+1 b0− b1 ) = ( b0 b0− b1 − 2b0− bs+2 b0− b1 +b0− bs+1 b0− b1 ) × 1. (7.7)
To determine λ2 we count the triples (Ω00, `, `0) in the order and its reversed
order, where Ω00 ∈ P2 satisfies (7.5), and `, `0 ⊆ Ω00 are lines containing x
such that ` − {x} ⊆ Ω − Ω0 and `0− {x} ⊆ Ω0 − Ω. We find
λ2× ( b0− bs+1 b0− b1 − b0− bs b0− b1 ) × ((b0− bs+1 b0− b1 − b0− bs b0− b1 ) = (b0 − bs+2 b0− b1 −b0− bs+1 b0− b1 ) × (b0− bs+2 b0− b1 − b0− bs+1 b0− b1 ) × 1. (7.8) (7.2) is immediate by solving (7.6)-(7.8) for a1(G).
Third we fix Ω00 ∈ P2 such that Ω ∩ Ω00 = ∆. We prove the number
c2(G) = c2(G)(Ω, Ω00) as expressed in (7.3). We do this by counting the
triples (Ω000, `, `00) in the order and its reversed order, where Ω000 ∈ P2 such
that Ω000∩ Ω, Ω000∩ Ω00 ∈ P
1, and `, `00 ⊆ Ω000 are lines containing x such that
` − {x} ⊆ Ω − Ω00 and `00− {x} ⊆ Ω00− Ω. We find c2(G) × ( b0− bs+1 b0− b1 − b0− bs b0− b1 ) × (b0− bs+1 b0 − b1 − b0− bs b0− b1 ) = (b0− bs+2 b0− b1 − b0 − bs b0− b1 ) × (b0 − bs+2 b0− b1 − b0− bs b0− b1 ) × 1. (7.9) (7.3) follows from (7.9).
Theorem 7.1 is a generalization of [4], which proves in the case ∆ = {x} for some vertex x of Γ and some additional assumptions.
Theorem 7.2. ([13, Theorem 8.6.33]). The complement G of an SRG(v, b0, a1, c2)
G is an SRG(v, v − b0− 1, v − 2b0+ c2− 2, v − 2b0+ a1)
Proof. For each adjacent pair u, w in G, there are 2(b0− 1) − a1 other vertices
in G1(u) ∪ G1(w), so u and w have v − 2 − 2(b0 − 1) + a1 = v − 2b0 + a1
common nonneighbors. When u, w are not adjacent, there are 2b0−c2vertices
Example 7.3. The Petersen graph G, an SRG (10, 3, 0, 1). Its complement G is an SRG (10, 6, 3, 4).
Next, we introduce the following lemma, which will be used later.
Lemma 7.4. Set
P := {∆0|∆0 ⊇ ∆ is a weak − geodetically closed subgraph of diameter s+1 in Γ}. Then |P | = bs bs− bs+1 . Proof. Observe |P | = ( bs b0− b1 )/(bs− bs+1 b0− b1 ) = bs bs− bs+1 by counting arguments. Lemma 7.5. Set
B = {∆00|∆00 ⊇ ∆ is a weak-geodetically closed subgraph of diameter s+2 in Γ}. Then |B| = bsbs+1
(bs+1− bs+2)(bs− bs+2)
.
Proof. We count the pair (∆0, ∆00) such that ∆ ∈ P, ∆0 ∈ B and ∆0 ⊆ ∆00 to
find |P | × ( bs+1 bs+1− bs+2 ) = |B| × bs− bs+2 (bs− bs+2) − (bs+1− bs+2) .
By Lemma 7.4 bs bs− bs+1 × ( bs+1 bs+1− bs+2 ) = |B| × bs− bs+2 bs− bs+1 . Hence |B| = bs bs− bs+1 × bs+1 bs+1− bs+2 ×bs− bs+1 bs− bs+2 = bsbs+1 (bs+1− bs+2)(bs− bs+2) .
Theorem 7.6. G(∆, 2) is a strongly regular graph with parameters
b0( ¯G) = bsb2s+2− b3s+2− b2s+1bs+2+ bs+1b2s+2 (bs− bs+1)(bs+1− bs+2)(bs− bs+2) . a1( ¯G) = 2b2 sb2s+1− bsb3s+1− 3b2sbs+1bs+2− 2bsb2s+1bs+2+ 2b3s+1bs+2− b3sbs+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) +5b 2 sb2s+2+ 3bsbs+1b2s+2− 2b2s+1b2s+2− 5bsb3s+2+ bs+1b3s+2+ b4s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) . c2( ¯G) = bsb3s+1+ 2b2sb2s+2− 3bsb3s+2− b2sbs+1bs+2+ 3bsbs+1b2s+2− bs+1b3s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) +−3bsbs+1b 2 s+2− bs+1b3s+2− 3bsb2s+1bs+2+ b2s+1+ b4s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) . Proof. Observe v = |P2| = |B| = bsbs+1 (bs+1− bs+2)(bs− bs+2)
and by Theorem 7.2, we have b0( ¯G) = v − b0(G) − 1 = bsbs+1 (bs+1− bs+2)(bs− bs+2) − bs+2(bs− bs+2) 2 (bs− bs+1)(bs+1− bs+2)(bs− bs+2) − (bs− bs+1)(bs+1− bs+2)(bs− bs+2) (bs− bs+1)(bs+1− bs+2)(bs− bs+2) = bsb 2 s+2− b3s+2− b2s+1bs+2+ bs+1b2s+2 (bs− bs+1)(bs+1− bs+2)(bs− bs+2) . a1( ¯G) = v − 2b0(G) + c2− 2 = bsbs+1 (bs+1− bs+2)(bs− bs+2) − 2 bs+2(bs− bs+2) 2 (bs− bs+1)(bs+1− bs+2)(bs− bs+2) +(bs− bs+2) 2 (bs− bs+1)2 − 2 = bsbs+1(bs− bs+1) 2 (bs+1− bs+2)(bs− bs+2)(bs− bs+1)2 −2 bs+2(bs− bs+2) 2(b s− bs+1) (bs+1− bs+2)(bs− bs+2)(bs− bs+1)2 + (bs− bs+2) 3(b s+1− bs+2) (bs+1− bs+2)(bs− bs+2)(bs− bs+1)2 −2(bs+1− bs+2)(bs− bs+2)(bs− bs+1) 2 (bs+1− bs+2)(bs− bs+2)(bs− bs+1)2 = 2b 2 sb2s+1− bsb3s+1− 3b2sbs+1bs+2− 2bsb2s+1bs+2+ 2b3s+1bs+2− b3sbs+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) +5b 2 sb2s+2+ 3bsbs+1b2s+2− 2b2s+1b2s+2− 5bsb3s+2+ bs+1b3s+2+ b4s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) .
c2( ¯G) = v − 2b0(G) + a1 = bsbs+1 (bs+1− bs+2)(bs− bs+2) − 2 bs+2(bs− bs+2) 2 (bs− bs+1)(bs+1− bs+2)(bs− bs+2) +(bs+1− bs+2 bs− bs+1 )2+ bs+2 bs+1− bs+2 − 1 = bsbs+1(bs− bs+1) 2 (bs+1− bs+2)(bs− bs+2)(bs− bs+1)2 − 2bs+2(bs− bs+2) 2(b s− bs+1) (bs+1− bs+2)(bs− bs+2)(bs− bs+1)2 + (bs+1− bs+2) 3(b s− bs+2) (bs+1− bs+2)(bs− bs+2)(bs− bs+1)2 + bs+2(bs− bs+1) 2(b s− bs+2) (bs+1− bs+2)(bs− bs+2)(bs− bs+1)2 −(bs+1− bs+2)(bs− bs+2)(bs− bs+1) 2 (bs+1− bs+2)(bs− bs+2)(bs− bs+1)2 = bsb 3 s+1+ 2b2sb2s+2− 3bsb3s+2− b2sbs+1bs+2+ 3bsbs+1b2s+2− bs+1b3s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) +−3bsbs+1b 2 s+2− bs+1b3s+2− 3bsb2s+1bs+2+ b2s+1+ b4s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) .
8
Another proof of Theorem 7.6
We prove Theorem 7.6 by a linear algebraic method in this section. Let Γ, ∆, P = P(∆), G(∆, i) be as in Section 7, and P, B be as in Lemma 7.4 and Lemma 7.5.
A quasisymmetric design (QSD) with parameters ρ < α is a 2 − (v, k, λ) design (such that |B ∩ B0| = ρ or α for all distinct blocks B and B0.)
From Lemma 7.4, we immediately have the following Lemma.
Lemma 8.1. (P, B) is a 2-(v, k, 1) quasisymmetric design with parameters 0, 1, where
v = bs/(bs− bs+1),
k = (bs− bs+2)/(bs− bs+1).
Corollary 8.2. (P, B) is a 2-(v, k, 1) quasisymmetric design with parameters 0, 1. Then (i) b = bsbs+1 (bs− bs+2)(bs+1− bs+2) . (ii) r = bs+1 bs+1− bs+2 . Proof.
(i) By Lemma 4.4. Hence
b = λv(v − 1) k(k − 1) = v(v − 1) k(k − 1) = v k × v − 1 k − 1 = bsbs+1 (bs− bs+2)(bs+1− bs+2) .
(ii) By Lemma 4.5. Hence r = λ(v − 1) k − 1 = v − 1 k − 1 = bs+1 bs+1− bs+2 .
Below we will proof the Theorem 7.6 by another method.
Theorem 8.3. G(∆, 2) is a quasisymmetric design (QSD) with parameters ρ = 0, α = 1, λ = 1. By Corollary 5.6, G(∆, 2) is a strongly regular graph with parameters (ii) b0(G) = bsb2s+2− b3s+2− b2s+1bs+2+ bs+1b2s+2 (bs− bs+1)(bs+1− bs+2)(bs− bs+2) . (iii) a1(G) = 2b2 sb2s+1− bsb3s+1− 3b2sbs+1bs+2− 2bsb2s+1bs+2+ 2b3s+1bs+2− b3sbs+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) +5b 2 sb2s+2+ 3bsbs+1b2s+2− 2b2s+1b2s+2− 5bsb3s+2+ bs+1b3s+2+ b4s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) . (iv) c2(G) = bsb3s+1+ 2b2sb2s+2− 3bsb3s+2− b2sbs+1bs+2+ 3bsbs+1b2s+2− bs+1b3s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) +−3bsbs+1b 2 s+2− bs+1b3s+2− 3bsb2s+1bs+2+ b2s+1+ b4s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) .
Proof. Where k = (bs− bs+2)/(bs− bs+1), b = bsbs+1/(bs− bs+2)(bs+1− bs+2),
r = bs+1/(bs+1− bs+2).
(ii) From Corollary 5.6, we have
b0(G) = −kr + k − 1 + b. Hence b0(G) = −kr + k − 1 + b = −(bs− bs+2)bs+1 (bs− bs+1)(bs+1− bs+2) + bs− bs+2 (bs− bs+1) − 1 + bsbs+1 (bs− bs+2)(bs+1− bs+2) = −bs+1(bs− bs+2) 2 (bs− bs+1)(bs+1− bs+2)(bs− bs+2) + (bs− bs+2) 2(b s+1− bs+2) (bs− bs+1)(bs+1− bs+2)(bs− bs+2) −(bs− bs+1)(bs+1− bs+2)(bs− bs+2) (bs− bs+1)(bs+1− bs+2)(bs− bs+2) + bsbs+1(bs− bs+1) (bs− bs+1)(bs+1− bs+2)(bs− bs+2) = bsb 2 s+2− b3s+2− b2s+1bs+2+ bs+1b2s+2 (bs− bs+1)(bs+1− bs+2)(bs− bs+2) .
(ii) From Corollary 5.6, we have
Hence a1(G) = 2k − 2kr − 2 + b + k2 = 2(bs− bs+2) (bs− bs+1) − 2(bs− bs+2)bs+1 (bs− bs+1)(bs+1− bs+2) − 2 + bsbs+1 (bs− bs+2)(bs+1− bs+2) +(bs− bs+2) 2 (bs− bs+1)2 = (2b 2 s+2− 2bsbs+1+ 2b2s+1− 2bs+1bs+2)(bs− bs+1)(bs− bs+2) (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) + bsbs+1(bs− bs+1) 2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) + (bs− bs+2) 3(b s+1− bs+2) (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) = 2b 2 sb2s+1− bsb3s+1− 3b2sbs+1bs+2− 2bsb2s+1bs+2+ 2b3s+1bs+2− b3sbs+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) +5b 2 sb2s+2+ 3bsbs+1b2s+2− 2b2s+1b2s+2− 5bsb3s+2+ bs+1b3s+2+ b4s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) .
(iv) From Corollary 5.6, we have
Hence c2(G) = r − 2kr + k2+ b − 1 = bs+1 (bs+1− bs+2) − 2 (bs− bs+2bs+1) (bs− bs+1)(bs+1− bs+2) +(bs− bs+2) 2 (bs− bs+1)2 + bsbs+1 (bs− bs+2)(bs+1− bs+2) − 1 = bs+1(bs− bs+1) 2(b s− bs+2) (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) − 2(bs− bs+1)(bs− bs+2) 2b s+1 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) + (bs− bs+2) 3(b s+1− bs+2) (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) + bsbs+1(bs− bs+1) 2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) −(bs− bs+1) 2(b s+1− bs+2)(bs− bs+2) (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) = bsb 3 s+1+ 2b2sb2s+2− 3bsb3s+2− b2sbs+1bs+2+ 3bsbs+1b2s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) + −bs+1b 3 s+2− bs+2)(bs− bs+2) (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) −3bsbs+1b2s+2− bs+1b3s+2− 3bsb2s+1bs+2+ b2s+1+ b4s+2 (bs− bs+1)2(bs+1− bs+2)(bs− bs+2) .
From the proof of Theorem 8.3, we obtain result similar to theorem 7.6.
References
[1] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cam-bridge University Press (Second Edition 2001).
[2] A. E. Brouwer and H. A. Wilbrink, The structure of near polygons with quads., Geom. Dedicata, 14(1883)145-176.
[3] J. Guo and S. Gao, Lattices Generated by Join of Strongly Closed Subgraphs in d-bounded Distance-regular Graphs, preprint.
[4] J. Guo and S. Gao, A Construction of a Distance-regular Graphs with Diameter 2, preprint.
[5] A. A. Ivanov and S. V. Shpectorov, Characterization of the association schemes of Hermitian forms over GF (22), Geom. Dedicata,
30(1989)23-33.
[6] A. Hiraki, Strongly closed subgraphs in a regular thick near polygon, European J. Combin., 20(8)(1999) 789-796.
[7] H. Suzuki, On strongly closed subgraphs of highly regular graphs, Eu-ropean J. Combin., 16(1995)197-220.
[8] H. Suzuki, Strongly closed subgraphs of a distance-regular graph with geometric girth five, Kyushu Journal of Mathematics, 50(2)(1996)371-384.
[9] E. E. Shult and A. Yanushka, Near n-gons and line systems, Geom. Dedicata, 9(1980)1-72.
[10] C. Weng, Weak-Geodetically Closed Subgraphs in Distance-Regular Graphs, Graphs and Combinatorics, 14(1998)275-304.
[11] C. Weng, D-bounded Distance-Regular Graphs, Europ. J. Comb., 18(1997)211-229.
[12] C. Weng, Classical Distance-Regular Graphs of Negative Type, J. Com-bin. Theory Ser. B, 76(1996)93-116.
[13] Douglas B. West. Introduction to Graph Theory, Prentice Hall , Second Edition 2001.