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

Abstract

)}. We give a necessary and sufficient condition for the existence of a strongly closed subgraph which is (c r +1 + a r +1 )-regular of diameter r + 1 containing a given pair of vertices at distance r + 1.

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

“…P Next we prove Theorem 1.3. For the cases r = 1 and q = r + 1 are proved in [8] and [10], respectively. So we may assume that r ≥ 2 and q ≥ r + 2.…”

“…In this paper, we generalize the argument we introduced in the previous paper [10] and give a sufficient condition for the existence of a strongly closed subgraph of diameter q containing a given pair of vertices at distance q. It is known that a strongly closed subgraph of a distance-regular graph is also distance-regular if it is regular.…”

“…Also we have shown some kinds of distance-regular graphs satisfying (C R). (See [9][10][11].) In particular, a distance-regular graph with c 2r +1 = 1 satisfies the condition (C R), and in this case the subgraph is the collinearity graph of a Moore geometry of diameter r + 1 with valency a r +1 + 1.…”

“…The special case q = r + 1 has already been studied in [10]. In this paper, we study the general case for the size q and show that these notions are closely related to the existence of a strongly closed subgraph of the diameter q.…”

“…In the previous paper [10], we introduced the condition (C R) and proved that it is a necessary and sufficient condition for the existence of a strongly closed subgraph of diameter r + 1. Also we have shown some kinds of distance-regular graphs satisfying (C R).…”