Triangle- and pentagon-free distance-regular graphs with an eigenvalue multiplicity equal to the valency

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(γ + 1), γ ∈ N, or a folded k-cube, k odd and k ≥ 7. This is a generalization of the results of Nomura [10] and Yamazaki [13], where they classified bipartite distance-regular graphs with an eigenvalue multiplicity k and showed that all such graphs are 2-homogeneous. We also classify bipartite almost 2-homogeneous distance-regular graphs with diameter d ≥ 4. In particular, we prove that such a graph is either 2-homogeneous (and thus classified by Nomura and Yamazaki), or a folded k-cube for k even, or a generalized 2d-gon with order (1, k − 1). CONTENTS