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 that Γ is 1 -homogeneous distance-regular with intersection number a 1 > 0 and diameter 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 is at least 2 , then b ⩾ 1 and one of the following (i)–(vi) holds: (i) Γ is a regular near 2 D -gon, (ii) Γ is a Johnson graph J ( 2 D , D ) , (iii) Γ is a halved ℓ -cube with ℓ ∈ { 2 D , 2 D + 1 } , (iv) Γ is a folded Johnson graph J ¯ ( 4 D , 2 D ) , (v) Γ is a folded halved 4 D -cube, (vi) the valency of Γ is bounded by a function of