Distance-regular Subgraphs in a Distance-regular Graph, VI

Abstract

In this paper we give a sufficient condition for the existence of a strongly closed subgraph which is (c q + a q )-regular of diameter q containing a given pair of vertices at distance q in a distance-regular graph. Moreover we show that a distance-regular graph with r = max{ j | (c j , a j , b j ) = (c 1 , a 1 , b 1 )}, b q−1 > b q and c q+r = 1 satisfies our sufficient condition.

Strongly Closed Subgraphs in a Regular Thick Near Polygon

“…We write (N P) <m holds if (N P) i holds for any 1 ≤ i < m. Our results are generalizations of the result of Brouwer and Wilbrink [3] and an application of the result of [6].…”

“…Suppose is a graph such that c i , a i and b i exist for all 0 ≤ i ≤ q. Then the results are proved by the same manner as in [6].…”

“…In Section 2, we recall the method and results introduced in the previous paper [6]. In Section 3, we collect several basic properties and show that regular near polygons are distanceregular.…”

“…In this section, we recall a constructing method of strongly closed subgraphs and the results obtained in the previous paper [6]. For the proofs and more detailed descriptions the reader is referred to [6].…”

“…On the other hand we gave a constructing method of strongly closed subgraphs in a distanceregular graph of arbitrary numerical girth [6].…”