無三角形且含五邊形之距離正則圖

Abstract

考慮一個具有Q-多項式性質的距離正則圖Γ，假設Γ的直徑D至少為3且其相交參數 a_1=0且a_2≠0，我們將證明下列(i)-(iii)是等價的： (i) Γ具有Q-多項式性質且不含長度為3的平行四邊形。 (ii) Γ具有Q-多項式性質且不含任何長度為i的平行四邊形，其中 。 (iii) Γ具有古典參數(D,b,α,β)，其中b,α,β是實數，且b<-1。 而當條件(i)-(iii) 成立時，我們證得Γ具有3-bounded性質。利用這個性質，我們可以證明其相交參數c_2等於1或2；且如果c_2=1，則 (b,α,β) = (-2, -2,((-2)^{D+1}-1)/3)。

Let Γ denote a distance-regular graph with Q-polynomial property. Assume the diameter D of Γ is at least 3 and the intersection numbers a_1=0 and a_2≠0. We show the following (i)-(iii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3. (ii) Γ is Q-polynomial and contains no parallelograms of any length i for 3≦i≦D. (iii) Γ has classical parameters (D,b,α,β)，for some real constants b,α,β with b<-1. When (i)-(iii) hold, we show that Γ has 3-bounded property. Using this property we prove that the intersection number c_2 is either 1 or 2, and if c_2=1 then (b,α,β)=(-2,-2,((-2)^{D+1}-1)/3).