A361447 -id:A361447 - cp's OEIS frontend

A352175 The number of Feynman graphs in phi^3 theory with 2n vertices, 2 external legs.

1, 5, 30, 186, 1276, 9828, 86279, 866474, 9924846, 128592118, 1864888539, 29950693288, 527584198445, 10109318656565, 209256249845854, 4651751087878667, 110501782280985273, 2792991694461152344, 74832356485576239136, 2118333127408342718683, 63169771935593153194107

Offset: 0

R. J. Mathar, Mar 07 2022

nonn

a(n) is the number of multigraphs with 2n unlabeled nodes of degree 3 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.
R. J. Mathar, Illustrations

Cf. A129427 (no external legs), A352173 (degree 4 case), A361447 (connected).

a(0) prepended and terms a(9) and beyond from Andrew Howroyd, Mar 10 2023

A361412 Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop), loops allowed.

Original entry on oeis.org

1, 3, 12, 67, 441, 3464, 31616, 331997, 3961462, 53105424, 791237787, 12978022526, 232407307054, 4511887729886, 94385418177277, 2116529900006321, 50646269987874834, 1288091152941695791, 34697173459041347465, 986800102740080746702, 29548269236430810895013

Offset: 0

Andrew Howroyd, Mar 12 2023

nonn

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

a(n) is also the number of connected multigraphs on 2*n+1 unlabeled nodes with one vertex of degree 2 and all others of degree 3, loops allowed.

			The a(1) = 3 multigraphs correspond to either a triple edge rooted on one of the edges or to a single edge with a loop at each end and rooted on either the edge or the loop.

A361135 is the 4-regular version.

Cf. A005967 (unrooted), A129427, A361446, A361447, A361448.

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.

cp's OEIS Frontend

A352175 The number of Feynman graphs in phi^3 theory with 2n vertices, 2 external legs.

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A361412 Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop), 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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