A112410 Number of connected simple graphs with n vertices, n+1 edges, and vertex degrees no more than 4.

0, 0, 0, 1, 5, 17, 56, 182, 573, 1792, 5533, 16977, 51652, 156291, 470069, 1407264, 4193977, 12451760, 36838994, 108656009, 319583578, 937634011, 2744720126, 8018165821, 23379886511, 68056985580, 197800670948, 574068309840, 1663907364480, 4816910618093, 13929036720057

Offset

1,5

Comments

Such graphs are also referred to (e.g., by Hendrickson & Parks) as carbon skeletons with two rings, or bicyclic skeletons, although actual number of simple cycles in such graphs can exceed 2 (e.g., in the example). - Andrey Zabolotskiy, Nov 24 2017

Terms computed with nauty agree at least to a(20) with those computed by formula and sequences A125669, A125670, A125671, A305132. - Andrew Howroyd, May 26 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 2 on page 103.

Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs.

Formula

a(n) = A125669(n) + A125670(n) + A125671(n) + A305132(n). - Andrew Howroyd, May 26 2018

Example

The only such graph for n = 4 is:
o-o
|/|
o-o

Prog

(nauty/bash)
for n in {4..15}; do geng -c -D4 ${n} $((n+1)):$((n+1)) -u; done # Andrey Zabolotskiy, Nov 24 2017

Crossrefs

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, this sequence, A112619, A112408, A112424, A112425, A112426, A112442.

Cf. A121941 (any number of edges), A006820 (2n edges).

Cf. A125669, A125670, A125671, A305132.

Sequence in context: A081495 A191645 A146240 * A146271 A145371 A347750

Adjacent sequences: A112407 A112408 A112409 * A112411 A112412 A112413

Keyword

nonn

Author

Jonathan Vos Post, Dec 08 2005

Extensions

Corrected offset and new name from Andrey Zabolotskiy, Nov 20 2017

a(20) corrected by Andrey Zabolotskiy and Andrew Howroyd, May 26 2018

Terms a(21) and beyond from Andrew Howroyd, May 26 2018

Status

approved