A note on distance-regular graphs with a small number of vertices compared to the valency

Abstract

In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given alpha > 2, there are finitely many distance-regular graphs Gamma with valency k, diameter D >= 3 and v vertices satisfying v < alpha k unless (D = 3 and Gamma is imprimitive) or (D = 4 and Gamma is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k >= 3, diameter D >= 3 and C-2 >= epsilon k for a given 0 < epsilon < 1 unless (D = 3 and Gamma is imprimitive) or (D = 4 and Gamma is antipodal and bipartite). Crown Copyright (C) 2013 Published by Elsevier Ltd. All rights reserved.