1,3

Comment from Brendan McKay, Oct 06, 2003: A generalized Moore graph is a regular graph of degree r where the counts of vertices at each distance from any vertex are 1, r, r(r-1), r(r-1)^2, r(r-1)^3, ... with the last distance having every other vertex. That is, all the levels are full except possibly the last which must have the rest. Alternatively, the girth is as great as the naive bound allows and the diameter is as little as the naive bound allows. Or, the average distance between pairs of vertices achieves the naive lower bound. As far as I know, it is an open problem if there are infinitely many generalized Moore graphs of each degree.

Comment from Brendan McKay, Oct 06, 2003: I have more terms of this sequence somewhere!

B. D. McKay, personal communication.

B. D. McKay and R. G. Stanton, The current status of the generalized Moore graph problem, pp. 21-31 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric Weisstein's World of Mathematics, Generalized Moore Graph

The counts are for graphs with 2, 4, 6, 8, ... nodes. In particular, there is a unique graph with 10 nodes.

Sequence in context: A106585 A057227 A162663 this_sequence A014243 A124839 A117046

Adjacent sequences: A005004 A005005 A005006 this_sequence A005008 A005009 A005010

nonn

N. J. A. Sloane (njas(AT)research.att.com).