A006924
Number of connected trivalent graphs with 2n nodes and girth exactly 4.
Original entry on oeis.org
0, 0, 0, 1, 2, 5, 20, 101, 743, 7350, 91763, 1344782, 22160335, 401278984, 7885687604, 166870266608, 3781101495300
Offset: 0
- CRC Handbook of Combinatorial Designs, 1996, p. 647.
- Gordon Royle, personal communication.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Definition corrected to include "connected", and "girth at least 4" minus "girth at least 5" formula provided by
Jason Kimberley, Dec 12 2009
A184944
Number of connected 4-regular simple graphs on n vertices with girth exactly 4.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 12, 31, 220, 1606, 16828, 193900, 2452818, 32670329, 456028472, 6636066091, 100135577616, 1582718909051
Offset: 0
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(8)=1 graph is the complete bipartite graph K_{4,4}.
4-regular simple graphs with girth exactly 4: this sequence (connected),
A185044 (disconnected),
A185144 (not necessarily connected).
Connected 4-regular simple graphs with girth exactly g:
A184943 (g=3), this sequence (g=4),
A184945 (g=5).
a(23) was appended by the author once
A033886(23) was known, Nov 03 2011
A184964
Number of connected 6-regular simple graphs on n vertices with girth exactly 4.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 9, 6, 267, 3727, 483012, 69823723, 14836130862
Offset: 0
a(0)=0 because even though the null graph (on zero vertices) is vacuously 6-regular and connected, since it is acyclic, it has infinite girth.
The a(12)=1 graph is the complete bipartite graph K_{6,6}.
Connected 6-regular simple graphs with girth at least g:
A006822 (g=3),
A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g:
A184963 (g=3), this sequence (g=4).
A184954
Number of connected 5-regular simple graphs on 2n vertices with girth exactly 4.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 1, 7, 388, 406824, 1125022325, 3813549359274
Offset: 0
Connected 5-regular simple graphs with girth at least g:
A006821 (g=3),
A058275 (g=4).
Connected 5-regular simple graphs with girth exactly g:
A184953 (g=3), this sequence (g=4),
A184955 (g=5).
A184973
Number of connected 7-regular simple graphs on 2n vertices with girth exactly 3.
Original entry on oeis.org
0, 0, 0, 0, 1, 5, 1547, 21609300, 733351105933
Offset: 0
a(0)=0 because even though the null graph (on zero vertices) is vacuously 7-regular and connected, since it is acyclic, it has infinite girth.
The a(4)=1 complete graph on 8 vertices is 7-regular; it has 28 edges and 56 triangles.
Connected 7-regular simple graphs with girth at least g:
A014377 (g=3),
A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: this sequence (g=3),
A184974 (g=4).
-
A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A014377 = A@014377;
A181153 = A@181153;
a[n_] := A014377[[n + 1]] - A181153[[n + 1]];
a /@ Range[0, 8] (* Jean-François Alcover, Jan 27 2020 *)
A184970
Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth exactly g.
Original entry on oeis.org
1, 5, 1547, 21609300, 1, 733351105933, 1
Offset: 4
1;
5;
1547;
21609300, 1;
733351105933, 1;
?, 8;
?, 741;
?, 2887493;
Connected 7-regular simple graphs with girth at least g:
A184971 (triangle); chosen g:
A014377 (g=3),
A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g: this sequence (triangle); chosen g:
A184973 (g=3),
A184974 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g:
A198303 (k=3),
A184940 (k=4),
A184950 (k=5),
A184960 (k=6), this sequence (k=7),
A184980 (k=8).
A184971
Irregular triangle C(n,g) counting the connected 7-regular simple graphs on 2n vertices with girth at least g.
Original entry on oeis.org
1, 5, 1547, 21609301, 1, 733351105934, 1
Offset: 4
1;
5;
1547;
21609301, 1;
733351105934, 1;
?, 8;
?, 741;
?, 2887493;
Connected 7-regular simple graphs with girth at least g: this sequence (triangle); chosen g:
A014377 (g=3),
A181153 (g=4).
Connected 7-regular simple graphs with girth exactly g:
A184970 (triangle); chosen g:
A184973 (g=3),
A184974 (g=4).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g:
A185131 (k=3),
A184941 (k=4),
A184951 (k=5),
A184961 (k=6), this sequence (k=7),
A184981 (k=8).
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