A Local Approach to 1-Homogeneous Graphs

Let Γ be a distance-regular graph with diameter d. For vertices x and y of Γ at distance i, 1 ≤ i ≤ d, we define the sets Ci(x, y) = Γi−1(x) ∩ Γ(y), Ai(x, y) = Γi(x) ∩ Γ(y) and Bi(x, y) = Γi+1(x) ∩ Γ(y). Then we say Γ has the CABj property, if the partition CABi(x, y) = {Ci(x, y), Ai(x, y), Bi(x, y)} of the local graph of y is equitable for each pair of vertices x and y of Γ at distance i ≤ j. We show that if Γ has the CABj property then the parameters of the equitable partitions CABi(x, y) do not depend on the choice of vertices x and y at distance i for all i ≤ j. The graph Γ has the CAB property if it has the CABd property. We show the equivalence of the CAB property and the 1-homogeneous property in a distance-regular graph with a1 6= 0. Finally, we classify the 1-homogeneous Terwilliger graphs with c2 ≥ 2.