Towards a classification of 1-homogeneous distance-regular graphs with positive intersection number a_1

Let Γ be a graph with diameter at least two. Then Γ is said to be 1-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices x and y in Γ, the distance partition of the vertex set of Γ with respect to both x and y is equitable, and the parameters corresponding to equitable partitions are independent of the choice of x and y. Assume Γ is 1-homogeneous distance-regular with intersection number a_1>0 and D⩾ 5. Define b=b_1/(θ_1+1), where b_1 is the intersection number and θ_1 is the second largest eigenvalue of Γ. We show that if intersection number c_2⩾ 2, then b⩾ 1 and one of the following (i)–(vi) holds: (i) Γ is a regular near 2D-gon, (ii) Γ is a Johnson graph J(2D,D), (iii) Γ is a halved ℓ-cube where ℓ∈{2D,2D+1}, (iv) Γ is a folded Johnson graph J̅(4D,2D), (v) Γ is a folded halved (4D)-cube, (vi) the valency of Γ is bounded by a function of b. Using this result, we characterize 1-homogeneous graphs with classical parameters and a_1>0, as well as tight distance-regular graphs.