A184945 Number of connected 4-regular simple graphs on n vertices with girth exactly 5.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 8, 131, 3917, 123859, 4131991, 132160607, 4018022149, 118369811959

Offset

0,21

Links

Table of n, a(n) for n=0..28.

Jan Goedgebeur and Jorik Jooken, Exhaustive generation of edge-girth-regular graphs, arXiv:2401.08271 [math.CO], 2024. See p. 12.

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Formula

a(n) = A058343(n) - A058348(n).

Example

a(0)=0 because even though the null graph (on zero vertices) is vacuously 4-regular and connected, since it is acyclic, it has infinite girth.
The a(19)=1 graph is the unique (4,5) cage: the Robertson graph (see also A159191). It has the following adjacency lists.
01 : 02 03 04 05
02 : 01 06 07 08
03 : 01 09 10 11
04 : 01 12 13 14
05 : 01 15 16 17
06 : 02 09 12 15
07 : 02 10 13 16
08 : 02 11 14 17
09 : 03 06 13 17
10 : 03 07 14 18
11 : 03 08 16 19
12 : 04 06 16 18
13 : 04 07 09 19
14 : 04 08 10 15
15 : 05 06 14 19
16 : 05 07 11 12
17 : 05 08 09 18
18 : 10 12 17 19
19 : 11 13 15 18

Crossrefs

4-regular simple graphs with girth exactly 5: this sequence (connected), A185045 (disconnected), A185145 (not necessarily connected).

Connected k-regular simple graphs with girth exactly 5: A006925 (k=3), this sequence (k=4), A184955 (k=5).

Connected 4-regular simple graphs with girth at least g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).

Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), this sequence (g=5).

Sequence in context: A178173 A058891 A274171 * A058343 A267407 A337296

Adjacent sequences: A184942 A184943 A184944 * A184946 A184947 A184948

Keyword

nonn,hard,more

Author

Jason Kimberley, Feb 14 2011

Status

approved