We classify triangleand pentagon-free distance-regular graphs with diameter d ≥ 2, valency k, and an eigenvalue multiplicity k. In particular, we prove that such a graph is isomorphic to a cycle, a k-cube, a complete bipartite graph minus a matching, a Hadamard graph, a distance-regular graph with intersection array {k, k − 1, k − c, c, 1; 1, c, k − c, k − 1, k}, where k = γ(γ + 3γ + 1), c = γ(...