The distance-regular graphs with valency k≥2, diameter D≥3 and k_D−1+k_D≤2k

Let Γ be a distance-regular graph with valency k and diameter D, and let x be a vertex of Γ. We denote by k_i(0≤i≤D) the number of vertices at distance i from x. In this paper, we try to quantify the difference between antipodal and non-antipodal distance-regular graphs. We will look at the sum k_D−1+k_D, and consider the situation where k_D−1+k_D≤2k. If Γ is an antipodal distance-regular graph, then k_D−1+k_D=k_D(k+1). It follows that either k_D=1 or the graph is non-antipodal. And for a non-antipodal distance-regular graph, it was known that k_D(k_D−1)≥k and k_D−1≥k both hold. So, this paper concerns on obtaining more detailed information on the number of vertices for a non-antipodal distance-regular graph. We first concentrate on the case where the diameter equals three. In this case, the condition k_D+k_D−1≤2k is equivalent to the condition that the number of vertices is at most 3k+1. And we extend this result to all diameters. We note that although the result of the diameter 3 case is a corollary of the result of all diameters, the main difficulty is the diameter 3 case, and that the diameter 3 case confirms the following conjecture: there is no primitive distance-regular graph with diameter 3 having the M-property.