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(1)國 立 交 通 大 學 應用數學系 碩 士 論 文 無三角形距離正則圖之研究 Triangle-free distance-regular graphs. 研 究 生:呂 明 欣 指導教授:翁 志 文. 教 授. 中 華 民 國 九 十 四 年 六 月.

(2) 無三角形距離正則圖之研究 Triangle-free distance-regular graphs 研 究 生:呂明欣. Student: Ming-hsing Lu. 指 導 教 授:翁志文 教授. Advisor: Dr. 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 2005 Hsinchu, Taiwan, Republic of China. 中 華 民 國 九 十 四 年 六 月.

(3) 無三角形距離正則圖之研究 研 究 生:呂明欣. 指導教授:翁志文. 教授. 國 立 交 通 大 學 應 用 數 學 系. 摘. 要. 令 Γ = ( X , R) 為一個直徑 d ≥ 3 的距離正則圖。對 2 ≤ i ≤ d ,一個長度 i 的平行四邊形,是指有一組 X 中的點 xyzu,且滿足 ∂ ( x, y ) = ∂ ( z , u ) = 1,. ∂ ( x, u ) = i,及 ∂ ( x, z ) = ∂ ( y, z ) = ∂ ( y, u ) = i − 1。假設 Γ 中,相交參數 a1 = 0, a 2 ≠ 0 。我們證明下列 (i)-(ii) 是等價的。(i) Γ 是 Q-polynomial,且不包含 長度為3的平行四邊形;(ii) Γ 具有古典參數。引用上述的結果,我們顯示了, 如果 Γ 具有古典參數且相交參數 a1 = 0 , a 2 ≠ 0 ,那麼對每一組 X 中的點 (v, w),若距離 ∂ (v, w) = 2,則存在一個 Γ 的強正則子圖 Ω 包含 v 及 w 。 並且,對 Ω 中的所有點 x , Ω 2 ( x) 的導出子圖是一個直徑最多為3的 a 2 正則連通圖。. 中 華 民 國 九 十 四 年 六 月. i.

(4) Triangle-free distance-regular graph Student : Ming-hsing Lu. Advisor : Dr. Chih-wen Weng. Department of Applied Mathematics National Chiao Tung University. Abstract Let Γ = ( X , R) denote a distance-regular graph with distance function ∂. and diameter d ≥ 3. For 2 ≤ i ≤ d , by a parallelogram of length i , we mean a 4-tuple xyzu of vertices in X such that ∂ ( x, y ) = ∂ ( z , u ) = 1 , ∂ ( x, u ) = i , and ∂ ( x, z ) = ∂ ( y, z ) = ∂ ( y, u ) = i − 1 . Suppose the intersection number a1 = 0 , a 2 ≠ 0 in Γ . We prove the following (i)-(ii) are equivalent. (i) Γ is. Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters. By applying the above result we show that if Γ has classical parameters and the intersection numbers a1 = 0 , a 2 ≠ 0 , then for each pair of vertices v, w ∈ X at distance ∂ (v, w) = 2 , there exists a strongly regular subgraph Ω of Γ containing v, w . Furthermore, for each vertex x ∈ Ω , the. subgraph induced on Ω 2 ( x) is an a 2 -regular connected graph with diameter at most 3.. ii.

(5) 致. 謝. 來到交大已經兩年了,這兩年來有許多美好的回憶,也很幸運能夠順利 的完成學業,一路走來,要感謝很多人。最感謝的是我的指導教授翁志文老 師,老師的耐心指導,以及做學問的態度,都讓我受益良多。對我生活上的 關心,更讓我銘記在心。 再來要謝謝交大應數組合組的黃光明老師、傅恆霖老師、黃大原老師以 及陳秋媛老師,每一位老師的課程都使我收穫豐富。謝謝同門的潘業忠學長 在研究上的建議,而博士班的飛黃學長、君逸學長、宏賓學長、棨丰學長在 我遇到問題時,總是提供許多幫助。與班上同學維展、亮詮、經凱、國元、 世謙、曲敏、惠蘭、宜諴、祐寧的相處時光,更是我珍惜的回憶。也謝謝交 大應數排球隊的每一位隊員,因為有大家ㄧ起努力,才能有許多屬於我們的 光榮時刻。 感謝我的母親以及家人,總是給我最多的關懷與支持,讓我在求學路上 沒有後顧之憂。更謝謝我的女友芹儀,在我高興時陪我ㄧ起分享,失落時給 我鼓勵。 要道謝的人太多了,還有喻培、元勳、芳婷…等。總之,謝謝大家陪我 走過這一段歷程。畢業了,因為人生無不散的筵席,但是,卻有常在心頭的 回憶。. iii.

(6) Contents. Abstract (in Chinese). i. Abstract (in English). ii. Acknowledgment. iii. Contents. iv. 1. 1. Introduction. 2 Preliminaries. 2. 3 The Main Theorem. 10. References. 16. iv.

(7) 1. Introduction. It is shown that a distance-regular graph with classical parameters has the Q-polynomial property [2, Theorem 8.4.1]. To describe the converse, let Γ denote a Q-polynomial distance-regular graph with diameter d ≥ 3. Brouwer, Cohen, Neumaier proved that if Γ is a near polygon and has intersection number a1 6= 0 then Γ has classical parameters [2, Theorem 8.5.1]. Weng proves the same result by loosing the near polygon assumption, but instead assuming that the graph Γ contains no kites of length 2 and no kites of length 3 [7, Lemma 2.4]. For the complement, Weng shows Γ has classical parameters in the assumptions that Γ has diameter d ≥ 4, intersection numbers a1 = 0, a2 6= 0, and Γ contains no parallelograms of length 3 and no parallelograms of length 4 [9, Theorem 2.11]. We generalize Weng’s result as following.. Theorem 1.1. Let Γ = (X, R) denote a distance-regular graph with diameter d ≥ 3 and intersection numbers a1 = 0, a2 6= 0. Then the following (i)-(ii) are equivalent. (i) Γ is Q-polynomial and Γ contains no parallelograms of length 3. (ii) Γ has classical parameters.. By the results in [4] and [10], Theorem 1.1 has the following corollary.. Corollary 1.2. Let Γ denote a distance-regular graph with classical parameters and intersection numbers a1 = 0, a2 6= 0. Then for each pair of vertices 1.

(8) v, w ∈ X at distance ∂(v, w) = 2, there exists a strongly regular subgraph Ω of Γ containing v, w with intersection numbers of Ω ai (Ω) = ai (Γ), ci (Ω) = ci (Γ), bi (Ω) = a2 (Γ) + c2 (Γ) − ai (Γ) − ci (Γ). for 0 ≤ i ≤ 2.. Applying Corollary 1.2, we have the following corollary.. Corollary 1.3. Let Ω be a strongly regular graph with a1 = 0, a2 6= 0. Then Ω2 (x) is an a2 -regular connected graph with diameter at most 3 for all x ∈ Ω.. 2. Preliminaries. Let Γ = (X, R) be a graph consisting of a finite non-empty set X of vertices, and a finite set R of unordered pairs of distinct vertices called edges. For each vertex x in a graph Γ, the number of edges incident to x is the valency of x. Two vertices associate with each edge are called the endpoints of the edge. If e = xy is an edge of Γ , then e is said to join the vertices x and y, and these vertices x and y are said to be adjacent. A path is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the list. A graph is connected if each pair of vertices belong to a path. The length of a path is the number of the edges in the path. The 2.

(9) distance of two vertices x and y in Γ is the length of the shortest path from x to y, denoted by ∂(x, y). The diameter of Γ is max{∂(x, y) | x, y ∈ X} For the rest of this section, we review some definitions and basic concepts of distance-regular graphs. See Bannai and Ito[1] or Terwilliger[6] for more background information. Throughout this thesis, Γ = (X, R) will denote a connected, graph with vertex set X, edge set R, path-length distance function ∂, and diameter d ≥ 3. Γ is said to be regular, if all vertices in Γ have the same valency. A kregular graph is a graph with valency k of each vertex of the graph. Γ is said to be a strongly regular graph srg(v, k, λ, µ), if Γ is k-regular with diameter 2 and has the following two properties: (i) For any two adjacent vertices x and y, there are exactly λ vertices adjacent to x and to y. (ii) For any two nonadjacent vertices x and y, there are exactly µ vertices adjacent to x and to y. Note that srg(v, k, λ, µ) is a distance-regular graph of diameter 2 with a1 = λ, c2 = µ, b0 = k. For a vertex x ∈ X and 0 ≤ i ≤ d, set Γi (x) = {y | ∂(x, y) = i}. Γ is said to be distance-regular whenever for all integers 0 ≤ h, i, j ≤ d, and all vertices x, y ∈ X with ∂(x, y) = h, the number phij =| {z ∈ X | z ∈ Γi (x) ∩ Γj (y)} | 3.

(10) is independent of x, y. The constants phij are known as the intersection numbers of Γ. For convenience, set ci := pi1 0 ≤ i ≤ d, bi := pi1. i+1. i−1. for 1 ≤ i ≤ d, ai := pi1. i. for. for 0 ≤ i ≤ d − 1, and put bd := 0, c0 := 0, k := b0 .. It is immediate from the definition that bi 6= 0 for 0 ≤ i ≤ d − 1, ci 6= 0 for 1 ≤ i ≤ d, and k = b 0 = ai + b i + c i. for 1 ≤ i ≤ d.. (2.1). Note that a1 6= 0 implies a2 6= 0. See Figure 1.. z. x. y. Figure 1: ∂(x, y) = 3. Either ∂(x, z) = 2 or ∂(z, y) = 2.. A distance-regular graph Γ is called bipartite whenever a1 = a2 = · · · = ad = 0. See Figure 2. Γ is called a generalized odd graph whenever a1 = a2 = · · · = ad−1 = 0, ad 6= 0. See Figure 3. From now on, we fix a distance-regular graph Γ with diameter d ≥ 3. For 0 ≤ h, i, j ≤ d let phij denote the intersection numbers of Γ. Let MatX (R) denote the algebra of all the matrices over the real number field with the rows and columns indexed by the elements of X. The distance matrices of Γ are the matrices A0 , A1 , · · · , Ad ∈ MatX (R), defined by the 4.

(11) Figure 2: A bipartite distance-regular graph. Figure 3: A generalized odd graph rule (Ai )xy.   1, if ∂(x, y) = i; =  0, if ∂(x, y) 6= i. for x, y ∈ X.. Then A0 = I,. (2.2) where J = all 10 s matrix,. A0 + A1 + · · · + Ad = J Ati = Ai Ai Aj =. d X. (2.3). for 0 ≤ i ≤ d,. (2.4). phij Ah. (2.5). for 0 ≤ i, j ≤ d,. h=0. Ai Aj = A j Ai. for 0 ≤ i, j ≤ d. 5. (2.6).

(12) Let M denote the subspace of MatX (R) spanned by A0 , A1 , . . . , Ad . Then M is a commutative subalgebra of MatX (R), and is known as the BoseMesner algebra of Γ. By [1, p59, p64], M has a second basis E0 , E1 , · · · , Ed such that E0 =| X |−1 J,. (2.7) for 0 ≤ i, j ≤ d,. Ei Ej = δij Ei E0 + E1 + · · · + Ed = I, Eit = Ei. (2.8) (2.9). for 0 ≤ i ≤ d.. (2.10). The E0 , E1 , · · · , Ed are known as the primitive idempotents of Γ, and E0 is known as the trivial idempotent. Let E denote any primitive idempotent of Γ. Then we have −1. E =| X |. d X. θi∗ Ai. (2.11). i=0. for some θ0∗ , θ1∗ , · · · , θd∗ ∈ R, called the dual eigenvalues associated with E. Let ◦ denote entry-wise multiplication in MatX (R). Then Ai ◦ Aj = δij Ai. for 0 ≤ i, j ≤ d,. so M is closed under ◦. Thus there exists qijk ∈ R 0 ≤ i, j, k ≤ d such that −1. Ei ◦ Ej =| X |. d X. qijk Ek. for 0 ≤ i, j ≤ d.. k=0. Γ is said to be Q-polynomial with respect to the given ordering E0 , E1 , · · · , Ed of the primitive idempotents, if for all integers h, i, j (0 ≤ h, i, j ≤ d), 6.

(13) qijh = 0 (resp. qijh 6= 0) whenever one of h, i, j is greater than (resp. equal to) the sum of the other two. Let E denote any primitive idempotent of Γ. Then Γ is said to be Q-polynomial with respect to E whenever there exists an ordering E0 , E1 = E, · · · , Ed of the primitive idempotents of Γ, with respect to which Γ is Q-polynomial. If Γ is Q-polynomial with respect to E, then the associated dual eigenvalues are distinct [5, p384]. It is shown that if Γ is Q-polynomial with a2 = 0, that Γ is a bipartite graph or a generalized odd graph. Set V = R|X| (column vectors), and view the coordinates of V as being indexed by X. Then the Bose-Mesner algebra M acts on V by left multiplication. We call V the standard module of Γ. For each vertex x ∈ X, set xˆ = (0, 0, · · · , 0, 1, 0, · · · , 0)t ,. (2.12). where the 1 is in coordinate x. Also, let h, i denote the dot product hu, vi = ut v. for u, v ∈ V.. (2.13). Then referring to the primitive idempotent E in (2.11), we compute from (2.10)-(2.13) that hE xˆ, yˆi =| X |−1 θi∗. for x, y ∈ X,. (2.14). where i = ∂(x, y). The following theorem about Q-polynomial is used in this thesis. Theorem 2.1. [6, Theorem 3.3] Let Γ be Q-polynomial with respect to E with the distinct associated dual eigenvalues θ0∗ , θ1∗ , . . . , θd∗ . Then the following (i)(ii) are equivalent. 7.

(14) (i) For all integers h, i, j(1 ≤ h ≤ d), (0 ≤ i, j ≤ d)and for all x, y ∈ X such that ∂(x, y) = h, X z∈X ∂(x,z)=i ∂(y,z)=j. Ez −. X. Ez =. z∈X ∂(x,z)=j ∂(y,z)=i. θ∗ h i pij ∗ θ0. − θj∗ (Ex − Ey). − θh∗. (2.15). (ii) ∗ ∗ ∗ θi−2 − θi−1 = σ(θi−3 − θi∗ ). (2.16). for appropriate σ ∈ R \ {0}. Γ is said to have classical parameters (d, b, α, β) whenever the diameter of Γ is d ≥ 3, and the intersection numbers of Γ satisfy     i−1 i 1+α for 0 ≤ i ≤ d, ci = 1 1       d i i bi = − β−α for 0 ≤ i ≤ d, 1 1 1. (2.17) (2.18). where   i := 1 + b + b2 + · · · + bi−1 . 1. (2.19). Γ is said to have classical parameters if Γ is has classical parameters (d, b, α, β) for some constants d, b, α, β. It is shown that a distance-regular graph with classical parameters has the Q-polynomial property [2, Theorem 8.4.1]. Terwilliger proves the following theorem.. Theorem 2.2. [6, Theorem 4.2] Let Γ denote a distance-regular with diam eter d ≥ 3. Choose b ∈ R \ {0. − 1}, and let be as in (2.19). Then the following (i)-(ii) are equivalent. 8.

(15) (i) Γ is Q-polynomial with associated dual eigenvalues θ0∗ , θ1∗ , . . . , θd∗ satisfying θi∗. −. θ0∗. =. (θ1∗. −.   i 1−i b . 1. θ0∗ ). (ii) Γ has classical parameters (d, b, α, β) for some real constants α, β. From Theorem 2.2, we have ∗ θi∗ − θi+1 = b−i (θ0∗ − θ1∗ ).. (2.20). Pick an integer 2 ≤ i ≤ d. By a parallelogram of length i in Γ, we mean a 4-tuple xyzw of vertices of X such that ∂(x, y) = ∂(z, w) = 1,. ∂(x, w) = i,. ∂(x, z) = ∂(y, z) = ∂(y, w) = i − 1. See Figure 4. y. i-1. 1. x. w. i-1. 1. z. i-1. Figure 4: A parallelogram of length i.. 9.

(16) 3. The Main Theorem. Lemma 3.1. Let Γ denote a Q-polynomial distance-regular graph with a1 = 0 and diameter d ≥ 3. Fix an integer i for 2 ≤ i ≤ d and three vertices x,y,z with ∂(y, x) = 1,. ∂(x, z) = i − 1,. ∂(y, z) = i.. Then si = si (x, y, z) = ai−1. ∗ ∗ (θ1∗ − θi∗ )(θi−1 − θ1∗ ) + (θ2∗ − θi∗ )(θ0∗ − θi−1 ) , ∗ ∗ ∗ ∗ (θ0 − θi−1 )(θi−1 − θi ). where si (x, y, z) =| Γi−1 (y) ∩ Γi−1 (x) ∩ Γ1 (z) | .. (3.1). Proof. Let `i (x, y, z) =| Γi−1 (y) ∩ Γi (x) ∩ Γ1 (z) | . Since w ∈ Γi−1 (y) ∩ Γ1 (z) implies w ∈ Γi−1 (x) ∪ Γi (x), we have si (x, y, z) + `i (x, y, z) = ai−1 . By (2.15) we also have X Ew − w∈X ∂(x,w)=i−1 ∂(z,w)=1. X. Ew = ai−1. w∈X ∂(x,w)=1 ∂(z,w)=i−1. ∗ θi−1 − θ1∗ (Ex − Ez). ∗ θ0∗ − θi−1. (3.2). (3.3). Taking the inner product of (3.3) with yˆ using(2.14), we obtain ∗ si (x, y, z)θi−1. +. `i (x, y, z)θi∗. −. ai−1 θ2∗. ∗ θi−1 − θ1∗ ∗ = ai−1 ∗ (θ1 − θi∗ ). ∗ θ0 − θi−1. (3.4). Solving si (x, y, z) by using (3.2) and (3.4) we get, ∗ ∗ (θ1∗ − θi∗ )(θi−1 − θ1∗ ) + (θ2∗ − θi∗ )(θ0∗ − θi−1 ) si (x, y, z) = ai−1 . ∗ ∗ (θ0∗ − θi−1 )(θi−1 − θi∗ ). 10. (3.5).

(17) From Lemma 3.1, si (x, y, z) is a constant for any vertices x, y, z with ∂(y, x) = 1, ∂(x, z) = i − 1, ∂(y, z) = i. We use si for this value. Note that si = 0 if and only if Γ contains no parallelogram of length i.. Lemma 3.2. Let Γ denote a distance-regular graph with classical parameters (d, b, α, β) and a1 = 0, a2 6= 0. Then b < −1.. Proof. From (2.1), (2.17), (2.18), and since a1 = 0, a2 6= 0, we have −α(b + 1)2 = a2 − (b + 1)a1 = a2 > 0.. (3.6). α < 0.. (3.7). Hence. By direct calculation from (2.17), we get (c2 − b)(b2 + b + 1) = c3 > 0.. (3.8). Since b is an integer and b 6= 0, −1[2, p.195], we have b2 + b + 1 > 0.. (3.9). c2 > b.. (3.10). α(1 + b) = c2 − b − 1 ≥ 0.. (3.11). Then from (3.8), implies. By using (2.17), (3.10), we get. Hence b < −1, by (3.7) and since b 6= −1.. 11.

(18) Theorem 3.3. Let Γ denote a Q-polynomial distance-regular with diameter d ≥ 3 and a1 = 0, a2 6= 0. Then with referring to definition in (3.1) the following (i)-(iii) are equivalent. (i) s3 = 0. (ii) si = 0, for 3 ≤ i ≤ d. (iii) Γ has classical parameter (d, b, α, β).. Proof. (ii)⇒(i) Clear. (iii)⇒(ii) From (2.20) we have, ∗ θi∗ − θi+1 = b−i (θ0∗ − θ1∗ ). for some b ∈ R\{0, −1}. Therefore, for 3 ≤ i ≤ d, (θ1∗ − θi∗ ) = (θ0∗ − θ1∗ )(b−1 + b−2 + · · · + bi−1 ),. (3.12). ∗ (θi−1 − θ1∗ ) = −(θ0∗ − θ1∗ )(b−1 + b−2 + · · · + bi−2 ),. (3.13). (θ2∗ − θi∗ ) = (θ0∗ − θ1∗ )(b−2 + b−3 + · · · + bi−1 ),. (3.14). and ∗ (θ0∗ − θi−1 ) = (θ0∗ − θ1∗ )(b0 + b−1 + · · · + bi−2 ).. (3.15). Evaluate (3.5) using (3.12), (3.13), (3.14), (3.15), we find si = 0 for 3 ≤ i ≤ d.. (i)⇒(iii) Suppose s3 = 0. Then by setting i = 3 in (3.5), (θ1∗ − θ3∗ )(θ2∗ − θ1∗ ) + (θ2∗ − θ3∗ )(θ0∗ − θ2∗ ) = 0. 12. (3.16).

(19) Set b :=. θ1∗ − θ0∗ . θ2∗ − θ1∗. (3.17). Then θ2∗ = θ0∗ +. (θ1∗ − θ0∗ )(b + 1) . b. (3.18). Eliminating θ2∗ , θ3∗ in (3.16) using (3.18) and (2.16),we have, −(θ1∗ − θ0∗ )2 (σb2 + σb + σ − b) = 0. σb2. (3.19). for appropriate σ ∈ R \ {0}. Note that θ1∗ 6= θ0∗ , hence (θ1∗ − θ0∗ )2 (σb2 + σb + σ − b) = 0, so σ −1 =. b2 + b + 1 . b. (3.20). From Theorem 2.2, to prove that Γ has classical parameter, it suffices to prove that θi∗. −. θ0∗. =. (θ1∗. −.   i 1−i b (0 ≤ i ≤ d). 1. θ0∗ ). (3.21). We prove (3.21) by induction on i. The case i = 0, 1 are trivial and case i = 2 is from (3.18). Now suppose i ≥ 3. Then (2.16) implies ∗ ∗ ∗ θi∗ = σ −1 (θi−1 − θi−2 ) + θi−3. (3.22). Evaluate (3.22) using (3.20) and the induction hypothesis, we find θi∗ − θ0∗ is as in (3.21). Therefore Γ has classical parameter.. Theorem 3.4. Let Γ = (X, R) denote a distance-regular graph with intersection numbers a1 = 0, a2 6= 0. Then the following (i)-(ii) are equivalent. 13.

(20) (i) Γ is Q-polynomial and Γ contains no parallelograms of length 3. (ii) Γ has classical parameters.. Proof. (i) ⇒ (ii) Suppose Γ is Q-polynomial and contains no parallelogram of length 3. Then s3 = 0. Hence Γ has classical parameters by Theorem 3.3.. (ii) ⇒ (i) Suppose Γ has classical parameters. Then Γ has Q-polynomial property[8, Theorem 8.4.1]. Then (i) holds by Theorem 3.3.. By the results in [4] and [10], we have the following corollary.. Corollary 3.5. Let Γ denote a distance-regular graph with classical parameters and intersection numbers a1 = 0, a2 6= 0. Then for each pair of vertices v, w ∈ X at distance ∂(v, w) = 2, there exists a strongly regular subgraph Ω of Γ containing v, w. The intersection numbers of Ω are ai (Ω) = ai (Γ), ci (Ω) = ci (Γ), bi (Ω) = a2 (Γ) + c2 (Γ) − ai (Γ) − ci (Γ) for 0 ≤ i ≤ 2.. Corollary 3.6. Let Ω be a strongly regular graph with a1 = 0, a2 6= 0. Then Ω2 (x) is an a2 -regular connected graph with diameter at most 3 for all x ∈ Ω.. 14.

(21) Proof. Fix a vertex x ∈ Ω, suppose y ∈ Ω2 (x), obviously, ∂(x, y) = 2. Hence | Ω1 (y) ∩ Ω2 (x) |= a2 . This shows Ω2 (x) is a2 -regular. Suppose that Ω2 (x) is not connected or is connected with diameter at least 4. Pick u, v ∈ Ω2 (x) such that there is no path in Ω2 (x) of length at most 3 connecting u, v. Observe ∂(u, v) = 2, since Ω has diameter 2. For each vertex z ∈ Ω1 (u) ∩ Ω1 (v), we must have ∂(x, z) = 1, otherwise ∂(x, z) = 2 and u, z, v is a path of length 2 in Ω2 (x). Hence we have z ∈ Ω1 (u) ∩ Ω1 (x) and Ω1 (u) ∩ Ω1 (v) ⊆ Ω1 (u) ∩ Ω1 (x). Now Ω1 (u) ∩ Ω1 (v) = Ω1 (u) ∩ Ω1 (x), since both sets have the same cardinality c2 . Similarly, we have Ω1 (u) ∩ Ω1 (v) = Ω1 (v) ∩ Ω1 (x). Pick w ∈ Ω1 (u) ∩ Ω2 (v). Then ∂(x, w) = 2, since w 6∈ Ω1 (u) ∩ Ω1 (v) = Ω1 (u) ∩ Ω1 (x). We do not have a path of length 2 in Ω2 (x) connecting w, v, otherwise we can extend this path to a path of length 3 in Ω2 (x) connecting u, v. By the same argument as above, we have Ω1 (w) ∩ Ω1 (v) = Ω1 (w) ∩ Ω1 (x) = Ω1 (v) ∩ Ω1 (x). Now we have Ω1 (u) ∩ Ω1 (v) = Ω1 (v) ∩ Ω1 (x) = Ω1 (w) ∩ Ω1 (v) Pick z ∈ Ω1 (u) ∩ Ω1 (v) = Ω1 (w) ∩ Ω1 (v). Then z, u, w forms a triangle, a contradiction with a1 = 0.. 15.

(22) References [1] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, London, 1984. [2] A. E. Brouwer, A. M. Cohen, and A. Neumaier,. Distance-Regular. Graphs, Springer-Verlag, Berlin, 1989. [3] G. Dickie, Twice Q-polynomial distance-regular graphs are thin. European Journal of Comb., 16:555–560, 1995. [4] H. Suzuki, Strongly closed subgraph of a distance-regular graph with geometric girth five, Kyushu J. Math, 50(2):371–384, 1996. [5] P. Terwilliger, The subconstituent algebra of an association scheme I, J. Alg. Combin., 1:363–388, 1992. [6] P. Terwilliger, A new inequality for distance-regular graphs, Discrete Math., 137:319–332, 1995. [7] C. Weng, Kite-Free P - and Q-Polynomial Schemes Graphs and Combinatorics, 11:201–207, 1995. [8] C. Weng, D-bounded Distance-regular Graphs, Europ. J. Combinatorics, 18:211–229, 1997. [9] C. Weng, Parallelogram-Free Distance-Regular Graphs, J. Combin. Theory Ser. B, 71(2):231–243, 1997. [10] C. Weng, Weak-Geodetically Closed Subgraphs in Distance-Regular Graphs, Graphs and Combinatorics, 14:275–304, 1998. 16.

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數據

Figure 2: A bipartite distance-regular graph
Figure 4: A parallelogram of length i.

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