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 a2 6= 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 a4 = 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: {2μ2 + μ, 2μ2 + μ− 1, μ2, μ, 1; 1, μ, μ2, 2μ2 + μ− 1, 2μ2 + μ}, μ ∈ N, is known. For μ = 1 the intersection array is uniquely realized by the dodecahedron. For μ 6= 1 we show that there are no distance-regular graphs with this intersection array. c © 2007 Elsevier Ltd. All rights reserved.