具封閉性質的距離正則圖之研究(I)

Abstract

圖 上三點 Gx, , 滿足 yz),(),(),(zxzyyx?=?+? 時，則稱此三點具測地性。 推廣這概念，圖 上三點 Gx, , 滿足 yz ?(x, y) + ?( y, z) ? ?(x, z) +1 時， 則稱此三點具弱測地性。所以 G 的點子集 O 中的任意具弱測地性三點 x, , 都滿足 yz x, z ?O ? y ?O 時，我們稱 O 是一弱測地閉包。當 中任兩點 Gx, 都包含於一直徑為 的正則弱測地閉包時，我們稱 具y),(yx?G封閉性。之前我們證明一直徑 、相交參數 3?D01≠a、 12 ≠ c 且不含任何長度的平行四邊形的距離正則圖必具封閉性，我們計畫去尋找更多此類圖。在一個具封閉性的 距離正則圖中的所有弱測地閉包會形成一部分有序集，而其有豐富的幾何結構。我們計畫探討此結構並找出適當公設來描述及刻畫它們。

A sequence x, , of vertices of is geodetic whenever yzG),(),(),(zxzyyx畝=畝+畝. Hence a sequence x, , of vertices of is weak-geodetic whenever . A vertex subset of is weak-geodetically closed if for any weak-geodetic sequence yzG1),(),(),(+畝.畝+畝zxzyyxOGx, , of G, yzOyOzx坥谲坥,. is -bounded if for any vertices GDx, of G, yx, are contained in a common regular weak-geodetically closed subgraph of diameter y),(yx畝. It was shown that if is distance-regular with diameter , intersection numbers G3.D01?a, and without parallelograms of any length, then is -bounded. We will find more distance-regular graphs to be -bounded. 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 that this poset is a ranked meet semi-lattice with lower semi-modular atomic intervals. We will find more properties of this poset. Furthermore we plan to find axioms of the poset.