A005638 -id:A005638 - cp's OEIS frontend

A361409 Number of bicolored cubic graphs on 2n unlabeled vertices with n vertices of each color.

1, 0, 1, 5, 66, 1071, 27606, 887305, 34583357, 1562797351, 80177945542, 4597212665432, 291214532031215, 20193430937073303, 1521240318892230748, 123711268485285686123, 10801367759750192440520, 1007762402877770768660697, 100058924666668698411972015, 10533938778032068908299390227, 1172080056205294525370971027435

Offset: 0

Andrew Howroyd, Mar 11 2023

nonn

Adjacent vertices may have the same color.

Central coefficients of A361361.

Cf. A005638, A361406, A361410, A361411.

A387146 Number of unlabeled biconnected cubic simple graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 2, 5, 18, 81, 480, 3874, 39866, 497818, 7187627, 116349635

Offset: 0

Hugo Pfoertner, Aug 21 2025

Combos, Generate graphs.
Brendan McKay and Adolfo Piperno, nauty and Traces.

Cf. A002851, A005638, A058378.

A386962 Number of equivalence classes of connected 3-regular graphs on 2n unlabeled nodes up to local complementation.

Original entry on oeis.org

0, 1, 2, 4, 15, 60

Offset: 1

Tristan Cam, Aug 11 2025

Number of equivalences classes of 3-regular graphs on 2n nodes up to a sequence of local complementation or isomorphism, also called orbits for the local equivalence relation.
a(n) is necessarily less than:
  A005638(n) (number of non-isomorphic, not necessarily connected 3-regular graphs);
  A002851(n) (number of non-isomophic connected 3-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 3-regular graphs with 6 nodes and they are not equivalent up to a sequence of local complementation, thus a(3) = 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. A002851, A005638, A090899, A156800.

cp's OEIS Frontend

A361409 Number of bicolored cubic graphs on 2n unlabeled vertices with n vertices of each color.

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A387146 Number of unlabeled biconnected cubic simple graphs with 2n nodes.

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A386962 Number of equivalence classes of connected 3-regular graphs on 2n unlabeled nodes up to local complementation.

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