|
%I #26 Sep 08 2022 08:45:54
%S 1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N Number of connected 2-regular simple graphs with n vertices.
%C All simple graphs have girth at least 3. Acyclic graphs have infinite girth.
%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>
%F a(0)=1; for 0<n<3 a(n)=0; for n>=3 , a(n)=1.
%F Proof: The null graph is vacuously 2-regular. There are no 2-regular simple graphs with 1 or 2 vertices. The n-cycle has girth n. QED.
%o (Magma) [1,0,0,1^^97];
%Y 2-regular simple graphs (with girth at least 3): this sequence (connected), A165652 (disconnected), A008483 (not necessarily connected).
%Y 2-regular connected: this sequence (simple graphs), A000012 (multigraphs with loops allowed).
%Y Connected regular simple graphs: A005177 (any degree), A068934 (triangular array), specified degree k: this sequence (k=2), A002851 (k=3), A006820 (k=4),A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
%Y Connected 2-regular simple graphs with girth at least g: this sequence (g=3), A185115 (g=4), A185115 (g=5), A185116 (g=6), A185117 (g=7), A185118 (g=8), A185119 (g=9).
%Y Connected 2-regular simple graphs with girth exactly g: A185013 (g=3), A185014 (g=4), A185015 (g=5), A185016 (g=6), A185017 (g=7), A185018 (g=8).
%K nonn,easy
%O 0
%A _Jason Kimberley_, Jan 05 2011
|
