A352174 -id:A352174 - cp's OEIS frontend

A361135 The number of unlabeled connected fairly 4-regular multigraphs of order n, loops allowed.

1, 3, 8, 30, 118, 548, 2790, 16029, 101353, 706572, 5375249, 44402094, 395734706, 3786401086, 38711834576, 421217184135, 4860174299186, 59278045511959, 762055884150141, 10299293881159294, 145994591873294780, 2165938721141964179, 33564939201581495090, 542344644703485899950, 9122110321170144880053

Offset: 1

R. J. Mathar, Mar 02 2023

Edges are undirected, vertices not labeled. "Fairly" means that each vertex has degree 4, but two of these edges do not connect to a second vertex; they are "fins" in CAD speak or "half-edges" in perturbation theory. The two fins may be attached to the same or to two different nodes. In the usual mathematical nomenclature these are connected graphs of order n+2 with two vertices of degree 1 and n vertices of degree 4, loops allowed.

H. Kleinert, A. Pelster, B. Kastening, and M. Bachmann, Recursive graphical construction of Feynman diagrams and their multiplicities in Phi^4 and Phi^2*A theory, Phys. Rev. E 62 (2) (2000), 1537 Table II.
R. J. Mathar, Illustrations

Cf. A085549 (4-regular), A352174 (assuming rooted external legs).

Terms a(7) and beyond from Andrew Howroyd, Mar 05 2023

A361446 Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop), loops allowed.

Original entry on oeis.org

1, 3, 16, 99, 717, 5964, 56701, 611750, 7432491, 100838222, 1514749135, 24989362186, 449429188211, 8754181791029, 183621843677724, 4126714250580949, 98932328702693666, 2520187379996442269, 67980528958530199837, 1935753445850303203221, 58025998739501873764826

Offset: 0

Andrew Howroyd, Mar 12 2023

nonn

a(0) = 1 by convention. Loops add two to the degree of a node.

A352174 is the 4-regular version.

Cf. A005967 (unrooted), A129427, A352175, A361412 (rooted at unoriented edge), A361447, A361448.

G.f.: B(x)/C(x) where B(x) is the g.f. of A352175 and C(x) is the g.f. of A129427.

A361448 Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop) whose removal does not disconnect the graph, loops allowed.

Original entry on oeis.org

1, 2, 10, 66, 511, 4536, 45519, 512661, 6436571, 89505875, 1369509795, 22908806774, 416408493351, 8178599551905, 172690849144538, 3902128758180500, 93970611848528998, 2402929936231885063, 65029668312580777779, 1856984518220396165657, 55803367549204703645086

Offset: 0

Andrew Howroyd, Mar 12 2023

nonn

a(0) = 1 by convention. Loops add two to the degree of a node.
Instead of a rooted edge, the graph can be considered to have a pair of external legs (or half-edges). The external legs add 1 to the degree of a node, but do not contribute to the connectivity of the graph.
The 4-regular version of this sequence is A352174 since removing a single edge from a connected even degree regular graph cannot disconnect the graph.

			a(2) = 10 = A361447(2) + 1 because there is one graph where the orientation of the rooted edge makes a difference:
    1       __
   /| \    |  |
   ||  3---4  |
   \| /    |__|
    2
The nodes are labeled 1,2,3,4. There is a double edge between nodes 1 and 2 and a loop at node 4. Roots at the edges (1,3) and (3,1) are considered different because orientation is considered. Roots at (1,3) and (2,3) are considered the same because the resulting graphs are isomorphic. Roots at (3,4) or (4,3) are disallowed because the removal of that edge would disconnect the graph.

Cf. A005967 (unrooted), A129427, A352174, A352175, A361412, A361446, A361447.

G.f.: B(x) - x*C(x)^2 where B(x) is the g.f. of A361446 and C(x) is the g.f. of A361412.

A352173 The number of Feynman graphs in phi^4 theory with n vertices, 2 external legs.

Original entry on oeis.org

1, 2, 7, 23, 85, 340, 1517, 7489, 41276, 252410, 1706071, 12660012, 102447112, 898081422, 8477941776, 85729296020, 924345402273, 10584325318278, 128259347448244, 1639694094741643, 22053783907891362, 311294619360437722, 4601020643330758040, 71063337073204684379, 1144820435086864897289

Offset: 0

R. J. Mathar, Mar 07 2022

nonn

The generating function of this is the product of the g.f. of the connected diagrams (A352174) by the g.f. of the vacuum diagrams (A129429, including a term x^0 for the empty graph): x + 2*x^2 + 7*x^3 + 23*x^4 + ... = (x + x^2 + 3*x^3 + 10*x^4 + ...) * (1 + x + 3*x^2 + 7*x^3 + 20*x^4 + ...). - R. J. Mathar, Mar 05 2023

a(n) is the number of multigraphs with n unlabeled nodes of degree 4 plus 2 noninterchangeable nodes of degree 1, loops allowed. - Andrew Howroyd, Mar 10 2023

R. de Mello Koch and S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007, App. D.

Cf. A352174 (connected), A129429 (0 ext. legs), A352175 (degree 3 case).

Offset corrected and a(13) and beyond from Andrew Howroyd, Mar 10 2023

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A361135 The number of unlabeled connected fairly 4-regular multigraphs of order n, loops allowed.

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A361446 Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop), loops allowed.

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A361448 Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop) whose removal does not disconnect the graph, loops allowed.

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A352173 The number of Feynman graphs in phi^4 theory with n vertices, 2 external legs.

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