Towards a classification of $1$-homogeneous distance-regular graphs with positive intersection number a₁

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 a₁>0 and D 5. Define b=b₁/(θ₁+1), where b₁ is the intersection number and θ₁ is the second largest eigenvalue of Γ. We show that if intersection number c₂ 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 -cube where ∈ \2D,2D+1\, (iv) Γ is a folded Johnson graph J̄(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 a₁>0, as well as tight distance-regular graphs.