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D-bounded Distance-regular Graphs

✍ Scribed by Chih-Wen Weng

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
314 KB
Volume
18
Category
Article
ISSN
0195-6698

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✦ Synopsis

Let ⌫ Ο­ ( X , R ) denote a distance-regular graph with diameter D Ρƒ 3 and distance function ␦ . A (vertex) subgraph ⌬ ' X is said to be weak -geodetically closed whenever for all x , y ⌬ and all z X ,

⌫ is said to be D -bounded whenever , for all x , y X , x and y are contained in a common regular weak-geodetically closed subgraph of diameter ␦ ( x , y ) . Assume that ⌫ is D -bounded .

Let P ( ⌫ ) denote the poset the elements of which are the weak-geodetically closed subgraphs of ⌫ with partial order by reverse inclusion . We obtain new inequalities for the intersection numbers of ⌫ ; equality is obtained in each of these inequalities if f the intervals in P ( ⌫ ) are modular . Moreover , we show this occurs if ⌫ has classical parameters and D Ρƒ 4 . We obtain the following corollary without assuming ⌫ to be D -bounded : C OROLLARY . Let ⌫ denote a distance -regular graph with classical parameters ( D , b , ␣ , ␀ ) and D Ρƒ 4 . Suppose that b Ο½ Οͺ 1 , and suppose the intersection numbers a 1 ΟΆ 0 and c 2 ΟΎ 1 . Then ␀ Ο­ ␣ 1 Ο© b D 1 Οͺ b .

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