Distance-regular graph with large a1 or c2

In this paper, we study distance-regular graphs $\mathrm{\Gamma }$ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of $\mathrm{\Gamma }$. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter three or a line graph.