An Application of a Construction Theory of Strongly Closed Subgraphs in a Distance-regular Graph

Abstract

Letbe a distance-regular graph with (where r ≥ 2 and c r +1 > 1. We prove that r = 2 except for the case a 1 = a r +1 = 0 and c r +1 = 2 by showing the existence of strongly closed subgraphs.

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

“…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.…”