A185145 -id:A185145 - cp's OEIS frontend

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

Jason Kimberley, Feb 14 2011

			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

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

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).

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

A186155 Rank of n^2 when {(1/8)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/8)i^3 after j^2 when (1/8)i^3=j^2. Complement of A186154.

Original entry on oeis.org

2, 5, 7, 9, 10, 12, 14, 15, 17, 19, 20, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 122, 123, 124, 126, 127, 128, 130

Offset: 1

Clark Kimberling, Feb 13 2011

nonn

See A186154.

Cf. A185145, A186154, A186152, A186153.

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

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A186155 Rank of n^2 when {(1/8)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/8)i^3 after j^2 when (1/8)i^3=j^2. Complement of A186154.

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