無三角形距離正則圖之研究

Abstract

在一個直徑d > 3的距離正則圖中，若相交參數a_1 = 0，a_2不等於0，我們證明下列 (i)-(ii) 是等價的。(i)此圖是Q-polynomial，且不包含長度為3的平行四邊形；(ii)此圖具有古典參數。引用上述的結果，我們顯示了，如果距離正則圖具有古典參數且相交參數a_1 = 0，a_2不等於0，那麼對每一組圖中的點(v,w) ，若距離(v,w)=2 ，則此圖存在一個強正則子圖包含v及w。並且，對強正則子圖中所有的點x，在強正則子圖中，所有與x距離為2的點的導出子圖是一個直徑最多為3的a_2-正則連通圖。

Let a distance-regular graph with diameter 3. Suppose the intersection number a_1 = 0，a_2 is not equal to 0, We prove the following (i)-(ii) are equivalent. (i)This graph is Q-polynomial and contains no parallelograms of length 3; (ii)This graph has classical parameters. By applying the above result we show that if a distance-regular graph has classical parameters and the intersection numbers a_1 = 0，a_2 is not equal to 0,then for each pair of vertices (v,w) at distance 2, there exists a strongly regular subgraph of the graph containing (v,w). Furthermore, for each vertex x in the strongly regular subgraph, the subgraph induced on all the vertices y which (x,y) at distance 2 in the strongly regular subgraphis is an a_2-regular connected graph with diameterat most 3.