cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Andrew Howroyd, Mar 12 2023

Keywords

Comments

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.

Examples

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

Crossrefs

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

Formula

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

Views

Author

Andrew Howroyd, Mar 12 2023

Keywords

Comments

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.

Examples

			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.

Crossrefs

A361135 is the 4-regular version.
Cf. A005967 (unrooted), A129427, A361446, A361447, A361448.

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

Views

Author

Andrew Howroyd, Mar 12 2023

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Formula

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.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.