A323389 The number of connected, unlabeled, undirected, edge-signed cubic graphs (admitting loops and multiedges) on 2n vertices where the degree of the first sign is 2 at each node.

1, 2, 5, 19, 88, 553, 4619, 49137, 646815, 10053183, 178725865, 3555840644, 78048875298, 1871066903575, 48617053973267, 1360733669185473, 40810827325698897, 1305690378666580997, 44387116312631271929, 1597768080980647428027, 60710507893875818581964

Offset: 0

Author

R. J. Mathar, Jan 13 2019

Keywords

• nonn

Comments

Obtained from the cubic graphs A005967 (connected undirected cubic graphs that may have loops and/or multiedges) by signing each edge with a plus or a minus such that two pluses and one minus meet at each vertex.

Links

• Richard J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019) Section II.
• R. J. Mathar, Chord diagrams with directed chords, vixra:2410.0145 (2024) Section Outlook (illustrations)

Crossrefs

Cf. A005967 (unsigned), A054499 (only one cycle of pluses), A170946 (directed plus-edges).

Programs

PARI

\\ See A339645 for combinatorial species functions.
cycleIndexSeries(n)={1+sLog(sCartProd(sExp(dihedralGroupSeries(n)), sExp(symGroupCycleIndex(2)*x^2 + O(x*x^n))))}
seq(n)={Vec(substpol(OgfSeries(cycleIndexSeries(2*n)), x^2, x))} \\ Andrew Howroyd, May 05 2023

Extensions

Terms a(6) and beyond from Andrew Howroyd, May 05 2023