Distance-Regular Graphs with Valency k , Diameter D > 3 and at Most Dk

. Let Gamma be a distance-regular graph with valency k and diameter D > 3. It has been shown that for a fixed real number alpha > 2, if Gamma has at most alpha k vertices, then there are only finitely many such graphs, except for the cases where (D = 3 and Gamma is imprimitive) and (D = 4 and Gamma is antipodal and bipartite). And there is a classification for alpha < 3. In this paper, we further study such distance-regular graphs for alpha > 3. Let beta > 3 be an integer, and let Gamma be a distance-regular graph with valency k, diameter D > 3 and at most beta k + 1 vertices. Note that if D > beta + 1, then Gamma must have at least beta k + 2 vertices. Thus, the assumption that Gamma has at most beta k + 1 vertices implies that D < beta. We focus on the case where D = beta and provide a classification of distance-regular graphs having at most Dk + 1 vertices.