行政院國家科學委員會補助專題研究計畫期中進度報告
計畫編號：NSC 96－2628－M－009－015－計畫類別： 個別型計畫
執行期間： 96 年 8 月 1 日至 97 年 7 月 31 日
計畫名稱: 具封閉性質的距離正則圖(1/3)
計畫主持人： 翁志文 (國立交通大學應用數學系)
weng@math.nctu.edu.tw 計畫參與人員： 博士生：潘業忠、黃喻培、黃皜文。 碩士生： 卜文強、曾雅 榕、汪政緯、陳雅玲、方仁洲、廖彥琳。97 年 5 月 22 日 一、 中文摘要 我們研究無三角形且有五邊形的具古典參 數之距離正則圖, 已得到主要的分類上的結 果, 此結果將成為博士生潘業忠的博士論文 主要部分. 另一方面在有平行四邊形的距離 正則圖上, 我們發現也能探討測地封閉距離 政則子圖的存在性, 這與之前的研究都建立 在無平行四邊形這假設完全不同. 不過想要 很廣泛證明這些子圖的存在性, 需要適當的 假設及嚴密的推理, 希望在暑假期間能完成 部份結果. 關鍵詞: 距離正則圖、 測地封閉子圖 二、 英文摘要
We obtain a major classification of a triangle-free distance-regular graph with classical parameters and a pentagon. This result will be included in Pan, Yeh-jong’s Ph.
D. thesis. We also find a way to study the geodetically closed subgraphs in a distance-regular graph which contains parallelogram.
Keywords: distance-regular graph,
geodetically closed subgraphs
三 、 緣由與目的
The D-bounded distance-regular graphs were introduced in 1997[7] by the project
investigator. This became an important concept in the classification of classical
distance-regular graphs of negative type in 1999[4]. There are many interesting geometric structures constructed from a D-bounded graph that need to be investigated. They also have applications to pooling designs [11], [12], [13]. Recently the project investigator notices that several authors devote themselves to the study of -bounded distance-regular graphs as results shown in [17], [18], [19], [20], [21]. This lures the project investigator going back to this line of study.
D
Recall that a sequence x , , of vertices of G is geodetic whenever
) , ( ) , ( ) , (x y +∂ y z =∂ x z ∂ . Hence a sequence x, , of vertices of is weak-geodetic whenever y z G 1 ) , ( ) , ( ) , ( +∂ ≤∂ + ∂ x y y z x z . A vertex subset O of is
weak-geodetically closed if for any
weak-geodetic sequence
G
x, , of ,
. A weak-geodetically closed subset is also called a strongly
closed subset in some literatures, e.g. [21].
is -bounded if for any vertices
y z G O y O z x, ∈ ⇒ ∈ G i x,
of with distance at most i ,
y G x,
are contained in a common regular weak-geodetically closed subgraph of diameter . It was shown in 1998[5] that if is distance-regular with diameter , intersection numbers
, and without parallelograms of length up to , then
is -bounded. For the case
y ) , (x y ∂ G 3 ≥ D 0 1 ≠ a c2 ≠1 1 + i G i a1 =0,
recently we show is 3-bounded under an additional assumption, -polynomial.
G
Q
Let denote a -bounded distance-regular graph, where is the diameter of . Putting all the weak-geodetically subgraphs together ordering by reversed inclusion we have a poset structure. It was shown in [7], this poset is a ranked meet semi-lattice with lower semi-modular atomic intervals. In [19], [20], the Mobius functions on intervals of the poset are investigated. The dual interval with a single vertex as the least element is studied in [17] and with more assumption, a strongly regular graph will be found from the set of rank 2 elements of the dual interval. The project
investigator believes there are many more properties in this poset to be found.
G D
D
G
四、 結果與討論
Let G be triangle-free distance-regular with classical parameters and a pentagon. We have shown that the graph G is 3-bounded. By using the 3-bounded property of G, we construct a ranked lattice of rank 3. We conclude from the study in this lattice that the intersection number c_2 of G is at most 2. The above line of study is based on the 3-bounded property which implies the graph G does not contain a parallelogram. We also find a way to study the existence of geodetically closed subgraphs in a distance-regular graph which does contain a parallelogram.
五、 計畫成果自評
The result about the classification is submitted to JCTB in title “ A note on triangle-free distance-regular graphs with a pentagon”. There are two cases remaining open in the above classification, one in c_2=2 and the other in c_2=1. Further study is necessary. There is another paper in title “A
Combinatorial Representation of Coxeter Graphs over a Field of Two Elements, submitted to JA” submitted to Journal of Algebra.
References
[1] Yeh-jong Pan and Chih-wen
Weng, Triangle-free Distance-regular Graphs, submitted to Journal of Algebraic
Combinatorics
[2] Terwilliger, P. and Weng, C., An inequality for regular near polygons, European Journal of Combinatorics 26, 227-235, 2005
[3] Terwilliger, P. and Weng, C.,
Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra, European J. Combin. 25, 287--298, 2004 [4] Weng, C., Classical Distance-Regular Graphs of Negative Type, Journal of Combinatorial Theory Series B, 76: 93-116, 1999
[5] Weng, C., Weak-geodetically Closed Subgroups in Distance-Regular Graphs, Graphs and Combinatorics, 14:
275-304, 1998
[6] Liang, Y. and Weng, C., Parallelogram-Free Distance-Regular Graphs, Journal of
Combinatorial Theory Series B, 231-243, 1997
[7] Weng, C., D-bounded Distance-Regular Graphs, European Journal of Comb. 18, 211-229, 1997
[8] Weng, C., Kite-Free P- and Q-Polynomial Schemes, Graphs and Combinatorics, 201-207, 1995
[9] Hau-wen Huang, Yu-pei Huang and
Chih-wen Weng, More on Pooling Spaces, submitted to Discrete Mathematics
[10] Ito T., Terwilliger P. and Weng C., The quantum algebra Uq(sl2) and its equitable presentation, Journal of Algebra 298 (2006) 284-301 .
[11] A. G. D'yachkov, Frank K. Hwang, A. J. Macula, P. A. Vilenkin and Chih-wen Weng, A Construction of Pooling Designs with Some Happy Surprises, Journal of Computational Biology, 12(8), 1127-1134, 2005
[12] Tayuan Huang and Chih-wen Weng, Pooling Spaces and Non-Adaptive Pooling Designs, Discrete Mathematics, 282(1-3), 163-169, 2004
[13] Tayuan Huang and Chih-wen Weng, A note on decoding of superimposed codes, Journal of Combinatorial Optimization, 7, 381-384, 2003
[14] Shih, C. and Weng. C., On the Templates Corresponding to Cycle-Symmetric
Connectivity in Celluar Neutral Networks, International Journal of Bifurcation and Chaos, 12:2957-2966, 2002
[15] Wei, P., Weng, C.,A Typical Vertex of a Tree , Discrete Math, 226:337-345, 2001 [16] Shih, C., Weng, C. Cycle-Symmetric Matrices and Convergent Neutral Networks, Physical D, 146:213-220,2000
[17] Suogang Gao and Jun Guo, A construction of a Distance-regular Graph with Diameter 2, submitted to G & C [18] Jun Guo and Suogagng Gao, Lattices Generated by Join of Strongly Closed Subgraphs in d-bounded Distance-regular Graphs, submitted to DM
[19] Jun Guo, Wen Liu, Suogang Gao, Some counting formulae of strongly closed subgraphs in distance-regular graphs, preprint
[20] Suogang Gao, Jun Guo, Wen Lin, Lattices Generated by Strongly Closed Subgraphs in d-bounded Distance-regular Graphs, preprint
[21] Hiroshi Suzuki, On Strongly Closed Subgraphs with Diameter Two and Q-Polynomial Property, preprint, 2004
