A372191 Number of unlabeled simple graphs covering n vertices with a unique undirected cycle of length > 2.

0, 0, 0, 1, 2, 6, 16, 43, 117, 319, 875, 2409, 6692, 18614, 52099, 146186, 411720, 1162295, 3289994, 9330913, 26517036, 75481622, 215201178, 614398459, 1756392061, 5026955216, 14403488345, 41311616835, 118601561506, 340795908579, 980078195995

Offset: 0

Author

Gus Wiseman, Apr 27 2024

Keywords

• nonn

Comments

An undirected cycle in a graph is a sequence of distinct vertices, up to rotation and reversal, such that there are edges between all consecutive elements, including the last and the first.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Gus Wiseman, The a(3) = 1 through a(6) = 16 covering graphs with a unique cycle.

Crossrefs

For no cycles we have A144958 (non-covering A005195), labeled A105784 (non-covering A001858).

Counting triangles instead of cycles gives A372174 (non-covering A372194), labeled A372171 (non-covering A372172).

The non-covering version is A236570, labeled A372193.

The labeled version is A372195, column k = 1 of A372175.

A002807 counts cycles in a complete graph.

A006125 counts graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

A372167 counts graphs by triangles, non-covering A372170.

A372173 counts unlabeled graphs by triangles (non-covering A263340).

A372176 counts labeled graphs by directed cycles.

Cf. A000055, A137916, A137917, A137918, A140637, A370167, A370169.

Formula

First differences of A236570.

Extensions

a(7) onwards from Andrew Howroyd, Jul 31 2024