A186729 - OEIS

%I #15 Jun 03 2023 09:30:27

%S 1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,19

%N Number of connected regular simple graphs on n vertices with girth at least 9.

%H Andries E. Brouwer, <a href="http://www.win.tue.nl/~aeb/graphs/cages/cages.html">Cages</a>

%H House of Graphs, <a href="https://houseofgraphs.org/meta-directory/cubic">Cubic graphs</a>

%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/C_k-reg_girth_ge_g_index">Index of sequences counting connected k-regular simple graphs with girth at least g</a>

%e The null graph is vacuously regular; there is one 0-regular simple graph with 1 vertex, and one 1-regular simple graph with 2 vertices;  each of those three graphs, being acyclic, has infinite girth.

%e The n-cycle is the connected 2-regular graph with girth n.

%e The (3,9)-cages have order 58 and there are 18 of them.

%Y Connected regular graphs of any degree with girth at least g: A005177 (g=3), A186724 (g=4), A186725 (g=5), A186726 (g=6), A186727 (g=7), A186728 (g=8), this sequence (g=9).

%Y Connected k-regular simple graphs with girth at least 9: this sequence (all k), A186719 (triangular array), A185119 (k=2).

%K nonn,hard,more

%O 0,59

%A _Jason Kimberley_, Oct 22 2011