A local approach to 1-homogeneous graphs
Let Gamma be a distance-regular graph with diameter d. For vertices x and y of Gamma at distance i, 1 less than or equal to i less than or equal to d, we define the sets C-i (x, y) = Gamma (i-1)(x) boolean AND Gamma>(*) over bar * (y), A(i)(x, y) = Gamma (i)(x) boolean AND Gamma>(*) over bar * (y) and B-i (x, y) = Gamma (i+1)(x) boolean AND Gamma>(*) over bar * (y). Then we say Gamma 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 pair of vertices x and y of Gamma at distance i less than or equal to j. We show that if Gamma has the CAB(j) property then the parameters of the equitable partitions CAB(i)(x, y) do not depend on the choice of vertices x and y at distance i for all i less than or equal to j. The graph Gamma has the CAB property if it has the CAB(d) property. We show the equivalence of the CAB property and the 1-homogeneous property in a distance-regular graph with a(1) not equal 0. Finally, we classify the 1-homogeneous Terwilliger graphs with c(2) greater than or equal to 2.
1-homogeneous distance-regular graphs equitable partitions locally strongly-regular