A Distance-Regular Graph with Strongly Closed Subgraphs
نویسنده
چکیده
Let be a distance-regular graph of diameter d, valency k and r := max{i | (ci , bi ) = (c1, b1)}. Let q be an integer with r + 1 ≤ q ≤ d − 1. In this paper we prove the following results: Theorem 1 Suppose 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 ≤ q and for any pair of vertices at distance i there exists a strongly closed subgraph of diameter i containing them. Theorem 2 If r ≥ 2, then c2r+3 = 1. As a corollary of Theorem 2 we have d ≤ k2(r + 1) if r ≥ 2.
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