A Local Approach to 1-Homogeneous Graphs

Abstract

Let \G be a distance-regular graph with diameter d. For vertices x and y of \G at distance i, 1\le i \le d, we define the sets C_i(x,y)=\G_{i-1}(x) \cap \G(y), A_i(x,y)=\G_{i} (x) \cap \G(y) and B_i(x,y)=\G_{i+1}(x) \cap \G(y). Then we say \G 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 \G at distance i. We show that in \Gamma with 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 \le j. The graph \G 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\ne 0. Finally, we classify the 1-homogeneous Terwilliger graphs with c_2\ge 2.