On triangle-free distance-regular graphs with an eigenvalue multiplicity equal to the valency

Let Γ be a triangle-free distance-regular graph with diameter d ≥ 3 , valency k ≥ 3 and intersection number a 2 ≠ 0 . Assume Γ has an eigenvalue with multiplicity k . We show that Γ is 1-homogeneous in the sense of Nomura when d = 3 or when d ≥ 4 and a 4 = 0 . In the latter case we prove that Γ is an antipodal cover of a strongly regular graph, which means that it has diameter 4 or 5. For d = 5 the following infinite family of feasible intersection arrays: { 2 μ 2 + μ , 2 μ 2 + μ − 1 , μ 2 , μ , 1 ; 1 , μ , μ 2 , 2 μ 2 + μ − 1 , 2 μ 2 + μ } , μ ∈ N , is known. For μ = 1 the intersection array is uniquely realized by the dodecahedron. For μ ≠ 1 we show that there are no distance-regular graphs with this intersection array.