A Local Approach to 1-Homogeneous Graphs

Abstract

Let Г bea distance-regular graph with diameter d. For vertices x and y of Г at distancei, 1 ≤ i ≤ d, we define the setsC _i(x,y) = Гi−1(x) ⋂ Г(y), A _i (x,y) = Г _i (x) ⋂ Г(y) and B _i (x,y) = Г _i+1(x) ⋂ Г(y).Then we say Г 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 Гat distance i ≤ j. We show that in Гwith the CAB_j property then the parameters ofthe equitable partitions CAB i(x,y) do not dependon the choice of vertices x and y atdistance i for all i ≤ j. The graphГ has the CAB property if it has the CAB _d property. We show the equivalence of the CAB property and the1-homogeneous property in a distance-regular graph with a ₁ ≠0. Finally, we classify the 1-homogeneous Terwilligergraphs with c ₂ Г 2.