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

Let ΓΓ be a triangle-free distance-regular graph with diameter d≥3d≥3, valency k≥3k≥3 and intersection number a2≠0a2≠0. Assume ΓΓ has an eigenvalue with multiplicity kk. We show that ΓΓ is 1-homogeneous in the sense of Nomura when d=3d=3 or when d≥4d≥4 and a4=0a4=0. In the latter case we prove that ΓΓ is an antipodal cover of a strongly regular graph, which means that it has diameter 4 or 5. For d=5d=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=1μ=1 the intersection array is uniquely realized by the dodecahedron. For μ≠1μ≠1 we show that there are no distance-regular graphs with this intersection array.