A Distance-Regular Graph with Strongly Closed Subgraphs

Let be a distance-regular graph of diameter d, valency k and r := maxi | (c_i,b_i) = (c₁,b₁). Let q be an integer with r + 1 q d – 1.In this paper we prove the following results:Theorem 1Suppose for any pair of vertices at distance q there exists a strongly closed subgraph of diameter q containing them. Then for any integer i with 1 i qand for any pair of vertices at distance i there exists a strongly closed subgraph of diameter i containing them.Theorem 2If r 2, thenc_2r+3 1.As a corollary of Theorem 2 we have d k²(r + 1) if r 2.