On the connectedness of the complement of a ball in distance-regular graphs

Abstract

An important property of strongly regular graphs is that the second subconstituent of any primitive strongly regular graph is always connected. Brouwer asked to what extent this statement can be generalized to distance-regular graphs. In this paper, we show that if gamma is any vertex of a distance-regular graph I" and t is the index where the standard sequence corresponding to the second largest eigenvalue of I" changes sign, then the subgraph induced by the vertices at distance at least t from gamma, is connected.