A Local Approach to 1-Homogeneous Graphs

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• 作者：Aleksandar Jurivsic and Jack Koolen
• 关键词：Distance ; regular graphs ; equitable partitions ; 1 ; homogeneous ; locally strongly ; regular
• 刊名：Designs ; Codes and Cryptography
• 出版时间：2000
• 出版年：2000
• 期刊代码：32_09251022
• 类别：bio
• 卷：21
• 期：1/3
• 页码：127-147
• 数据来源：sp

摘要

Let bea distance-regular graph with diameter d. For vertices x and y of at distancei, 1 i d, we define the setsC_i(x,y) = i–1(x) (y), A_i(x,y) = _i(x) (y) and B_i(x,y) = _i+1(x) (y).Then we say has the CAB_j property,if the partition CAB_i(x,y) = {C_i(x,y),A_i(x,y),B_i(x,y)}of the local graph of y is equitable for each pairof vertices x and y of at distance i j. We show that in with the CAB_j property then the parameters ofthe equitable partitions CABi(x,y) do not dependon the choice of vertices x and y atdistance i for all i j. The graph has the CAB property if it has the CAB_dproperty. We show the equivalence of the CAB property and the1-homogeneous property in a distance-regular graph with a₁ 0. Finally, we classify the 1-homogeneous Terwilligergraphs with c₂ 2.