A361135 -id:A361135 - cp's OEIS frontend

A352174 The number of connected Feynman graphs in phi^4 theory with n vertices, 2 external legs.

1, 1, 3, 10, 39, 174, 853, 4632, 27607, 180148, 1281437, 9896652, 82610706, 742066238, 7142537105, 73372890090, 801485878718, 9278500053475, 113490189831585, 1462634417672439, 19811476725948185, 281387198719498556, 4182019967162071868, 64912032433421243992, 1050391945321019422595

Offset: 0

R. J. Mathar, Mar 07 2022

nonn

It appears (?) these are obtained from the undirected graphs of A361135 by counting graphs twice (or more often) if entering the graphs via the two external legs (marking these as an in- and an out-leg, alternatively considering their fins directed) makes a difference. There are e.g. 2 (out of 8) graphs on 3 vertices in A361135 that are not left-right-symmetric, and 9 (out of 30) on 4 vertices in A361135 which are not left-right-symmetric: a(4) = A361135(3)+2, a(5) = A361135(4)+9 (?). The index shift needed might be some sort of virtually connecting the two fins (half-edges) and considering that one more vertex. - R. J. Mathar, Mar 05 2023

a(n) is the number of connected 4-regular multigraphs on n unlabeled nodes rooted at an oriented edge, loops allowed. A361135(n) is the case for an unoriented edge. The term a(0)=1 is an artifact arising from the way the sequence was enumerated using a pair of vertices of degree 1 (see A352173). - 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. A352173 (not necessarily connected), A361135.

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

A361447 Number of connected 3-regular (cubic) multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop) whose removal does not disconnect the graph, loops allowed.

Original entry on oeis.org

1, 2, 9, 49, 338, 2744, 26025, 282419, 3463502, 47439030, 718618117, 11937743088, 215896959624, 4224096594516, 88919920910684, 2004237153640098, 48165411560792500, 1229462431057436457, 33221743136066636436, 947415638925100675208, 28436953641282225835143

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 A361135 since removing a single edge from a connected even degree regular graph cannot disconnect the graph.

			The illustrations in A352175 by _R. J. Mathar_ show 1, 2, 9, and 49 connected graphs corresponding to the initial terms of this sequence.

Cf. A005967 (unrooted), A129427, A352175, A361135, A361412, A361446, A361448.

G.f.: B(x) - x*(B(x)^2 + B(x^2))/2 where B(x) is the g.f. of A361412.

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.

A361698 The number of unlabeled connected 4 regular multigraphs on n nodes with 4 external legs, loops allowed.

Original entry on oeis.org

1, 1, 2, 8, 37, 181, 1010, 6135, 40893, 295753, 2317683, 19568427, 177397551, 1719790643, 17767328745, 194954224643, 2265042428226, 27785727158182, 358952560098959, 4871697965709175, 69309502018430799, 1031550920679805502, 16030923441853969843, 259682356008358417321, 4377679648827121988375

Offset: 0

R. J. Mathar, Mar 21 2023

nonn

Alternatively, unlabeled connected multigraphs on n+4 nodes with 4 nodes of degree 1 and n nodes of degree 4.

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 eq. (4.29)

Cf. A361135 (2 legs), A085549 (no legs), A129429, A361454 (not necessarily connected).

G.f.: 1 + B(x)/(1 + C(x)) - (D(x)^2 + D(x^2))/2 where B(x), C(x) and D(x) are the g.f.s of A361454, A129429 and A361135, respectively. - Andrew Howroyd, Mar 21 2023

Terms a(7) and beyond from Andrew Howroyd, Mar 21 2023

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A352174 The number of connected Feynman graphs in phi^4 theory with n vertices, 2 external legs.

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A361447 Number of connected 3-regular (cubic) multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop) whose removal does not disconnect the graph, loops allowed.

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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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A361698 The number of unlabeled connected 4 regular multigraphs on n nodes with 4 external legs, loops allowed.

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