A033301 -id:A033301 - cp's OEIS frontend

A180260 Number of not necessarily connected 8-regular simple graphs on n vertices.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 423187422492342, 281341168330848874, 214755319657939505396, 187549729101764460261505, 186685399408147545744203915, 210977245260028917322933165888

Offset: 0

Jason Kimberley, Jan 17 2011

The Euler transformation currently does nothing: for n < 18, a(n) = A014378(n).

			The a(0)=1 graph is K_0 (vacuously 8-regular).
The a(9)=1 graph is K_9.

Eric Weisstein's World of Mathematics, Octic Graph

8-regular simple graphs: A014378 (connected), A165878 (disconnected), this sequence (not necessarily connected).
Not necessarily connected regular simple graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A000012 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), this sequence (k=8).
8-regular not necessarily connected graphs: this sequence (simple graphs), A129437 (multigraphs with loops allowed), A129426 (multigraphs with loops forbidden).

A014378 = Cases[Import["https://oeis.org/A014378/b014378.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014378] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A014378.

a(17)-a(22) from Andrew Howroyd, Mar 08 2020

A005816 Number of 4-valent labeled graphs with n nodes where multiple edges and loops are allowed.

Original entry on oeis.org

1, 1, 3, 15, 138, 2021, 43581, 1295493, 50752145, 2533755933, 157055247261, 11836611005031, 1066129321651668, 113117849882149725, 13965580274228976213, 1985189312618723797371, 321932406123733248625851, 59079829666712346141491403, 12182062872168618012045410805

Offset: 0

Simon Plouffe

nonn

Each loop contributes 2 to the valency of its node.

Goulden, I. P.; Jackson, D. M.; Reilly, J. W.; The Hammond series of a symmetric function and its application to P-recursiveness. SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..25 from Jason Kimberley)
R. C. Read, The enumeration of locally restricted graphs (I), J. London Math. Soc. 34 (1959) 417-436.

Column k=4 of A333467.
Cf. A005815.
Cf. A129429 (unlabeled), A033301.

a(n) = N{E_n[S_4] * S_{2n}[S_2]}.

Definition corrected by appending "where multiple edges and loops are allowed", reference to Read 1959, formula from Read 1959 (5.11), and new terms a(16), a(17), a(18) contributed by Jason Kimberley, Jan 22 2010

A184326 The number of disconnected k-regular simple graphs on 2k+6 vertices.

Original entry on oeis.org

1, 1, 4, 9, 25, 66, 297, 1562, 10901, 88238, 806174, 8037887, 86228020, 985884104, 11946634677, 152808994328, 2056701656260

Offset: 0

Jason Kimberley, Jan 15 2011

			The a(0)=1 graph is 6K_1. The a(1)=1 graph is 4K_2. The a(2)=4 graphs are 2C_3+C_4, 2C_5, C_4+C_6, and C_3+C_7.

This sequence is the fifth highest diagonal of D=A068933: that is a(n)=D(2k+6, k).

Cf. A184324(k) = D(2k+4, k) and A184325(k) = D(4k+5, 2k).

a(0)=1, a(1)=1, a(2)=4, a(3)=9. For n>3, a(n) = A033301(k+5) + ((k+1)mod 2)*A005638(k div 2 + 2) + A000217(A008483(k+3)).

Proof: Let C=A068934, D=A068933, and E=A051031. Now a(n) = D(2k+6,k) = C(k+1,k)C(k+5,k) + C(k+2,k)C(k+4,k) + A000217(C(k+3,k)), from the disconnected Euler transform. Notice that D(k+i,k)=0 provided k+i < 2k+2; that is k > i-2. So if i <= 5 and k > 3, then D(k+i,k)=0. Hence for k > 3, a(n) = E(k+1,k)E(k+5,k) + E(k+2,k)E(k+4,k) + A000217(E(k+3,k)) = E(k+1,0)E(k+5,4) + E(k+2,1)E(k+4,3) + A000217(E(k+3,2)). We have E(k+1,0)=1, and E(k+2,1)=(k+1)mod 2. For even k, E(k+4,3)=A005638(k div 2 + 2); for odd k, E(k+2,1)=0. QED.

A185140 Irregular triangle E(n,g) counting not necessarily connected 4-regular simple graphs on n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 2, 5, 1, 16, 0, 58, 2, 264, 2, 1535, 12, 10755, 31, 87973, 220, 803973, 1606, 8020967, 16829, 86029760, 193900, 983431053, 2452820, 11913921910, 32670331, 1, 152352965278, 456028487, 2, 2050065073002, 6636066126, 8, 28466234288520, 100135577863, 131, 8020967, 16829

Offset: 5

Jason Kimberley, Jan 06 2013

The first column is for girth at least 3. The column for girth g commences when n reaches A037233(g).

			05: 1;
06: 1;
07: 2;
08: 5, 1;
09: 16, 0;
10: 58, 2;
11: 264, 2;
12: 1535, 12;
13: 10755, 31;
14: 87973, 220;
15: 803973, 1606;
16: 8020967, 16829;
17: 86029760, 193900;
18: 983431053, 2452820;
19: 11913921910, 32670331, 1;
20: 152352965278, 456028487, 2;
21: 2050065073002, 6636066126, 8;
22: 28466234288520, 100135577863, 131;

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Initial columns of this triangle: A185143 (g=3), A185144 (g=4).

The n-th row is the sequence of differences of the n-th row of A185340:
E(n,g) = A185340(n,g) - A185340(n,g+1), once we have appended 0 to each row of A185340.
Hence the sum of the n-th row is A185340(n,3) = A033301(n).

A033700 Number of connected transitive 4-valent (or quartic) graphs with n nodes.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 2, 10, 3, 5, 7, 13, 4

Offset: 5

Ronald C. Read

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

Cf. A006820, A033301, A033483.

a(16)-a(17) from Sean A. Irvine, Jul 13 2020

A173758 Partial sums of A006820.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 5, 11, 27, 86, 351, 1895, 12673, 100841, 906332, 8943750, 95165384, 1081035906, 13027523553, 165835586734, 2222527601208, 31273800434817, 460941981112256, 7101107185967292, 114127691657536897, 1910229280483131905, 33244227211086415436

Offset: 0

Jonathan Vos Post, Feb 23 2010

nonn

Number of connected quartic graphs with at most n nodes.

Cf. A006820, A033301, A033483.

Connected regular graphs of degree k: 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).

a(n) = Sum_{k=0..n} A006820(k). [corrected by Georg Fischer, Sep 28 2021]

Corrected by Jason Kimberley, Mar 29 2010

A385629 Number of equivalence classes of connected 4-regular graphs on n unlabeled nodes up to local complementation.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 6, 13, 56, 261

Offset: 1

Tristan Cam, Aug 09 2025

Number of equivalences classes of 4-regular graphs on n nodes up to a sequence of local complementation or isomorphism.
a(n) is necessarily less than:
  A033301(n) (number of non-isomorphic, not necessarily connected 4-regular graphs);
  A006820(n) (number of non-isomophic connected 4-regular graphs);
  A090899(n) (number of local equivalence classes of connected graphs); and
  A156800(n) (number of equivalence classes for connected graphs up to pivots and isomorphism).
This is relevant in the study of optimal quantum circuit synthesis for graph state preparation.

			There are only two 4-regular graphs with 7 nodes and they are not equivalent up to a sequence of local complementation, thus a(7) = 2.

Niels Bohr Institute Center for Hybrid Quantum Networks, graph_table (github)
Tristan Cam, Cyril Gavoille, Yvan Le Borgne, and Simon Martiel, Universal Graph Theory Operations for Graph State Preparation

Cf. A006820, A033301, A090899, A156800.

cp's OEIS Frontend

A180260 Number of not necessarily connected 8-regular simple graphs on n vertices.

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A005816 Number of 4-valent labeled graphs with n nodes where multiple edges and loops are allowed.

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A184326 The number of disconnected k-regular simple graphs on 2k+6 vertices.

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A185140 Irregular triangle E(n,g) counting not necessarily connected 4-regular simple graphs on n vertices with girth exactly g.

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A033700 Number of connected transitive 4-valent (or quartic) graphs with n nodes.

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A173758 Partial sums of A006820.

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A385629 Number of equivalence classes of connected 4-regular graphs on n unlabeled nodes up to local complementation.

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