A141580 Number of unlabeled non-mating graphs with n vertices.

0, 1, 2, 6, 18, 78, 456, 4299, 68754, 1990286, 106088988, 10454883132, 1904236651216, 641859005526860, 401547534010157680, 467956331904669136874, 1019785644052109276678788, 4171197546082606538129623140

Offset: 1

Author

Tanya Khovanova, Aug 19 2008

Keywords

• nonn

Comments

a(n) is the difference between A000088 (number of graphs on n unlabeled nodes) and A004110 (number of n-node graphs without endpoints)

A non-mating graph has two vertices with an identical set of neighbors.

The adjacency matrix of a non-mating graph is degenerate.

Also the number of unlabeled graphs with n vertices and at least one endpoint. - Gus Wiseman, Sep 11 2019

Examples

			A cycle with 4 vertices is a non-mating graph. In the standard ordering of vertices, vertices 1 and 3 are both connected to vertices 2 an 4, thus having an identical sets of neighbors.
From _Gus Wiseman_, Sep 11 2019: (Start)
Non-isomorphic representatives of the a(2) = 1 through a(5) non-mating graph edge-sets:
  {12}  {12}     {12}           {12}
        {13,23}  {12,34}        {12,34}
                 {13,23}        {13,23}
                 {13,24,34}     {12,35,45}
                 {14,24,34}     {13,24,34}
                 {14,23,24,34}  {14,24,34}
                                {12,34,35,45}
                                {13,24,35,45}
                                {14,23,24,34}
                                {14,25,35,45}
                                {15,25,35,45}
                                {12,25,34,35,45}
                                {14,25,34,35,45}
                                {15,23,24,35,45}
                                {15,25,34,35,45}
                                {13,24,25,34,35,45}
                                {15,24,25,34,35,45}
                                {15,23,24,25,34,35,45}
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Ronald C. Read, The enumeration of mating-type graphs, Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
• Gus Wiseman, The a(1) = 1 through a(5) = 18 non-mating graphs (isolated vertices not shown).
• Gus Wiseman, The a(1) = 1 through a(5) = 18 graphs with at least one endpoint (isolated vertices not shown).

Crossrefs

The labeled version is A327379.

Cf. A000088, A004110, A028242, A059167, A245797, A327335, A327371.

Programs

Mathematica

k = {}; For[i = 1, i < 8, i++, lg = ListGraphs[i] ; len = Length[lg]; k = Append[k, Length[Select[Range[len], Length[Union[ToAdjacencyMatrix[lg[[ # ]]]]] != i &]]]]; k

Formula

a(n) = A000088(n) - A004110(n).

Extensions

Extended by R. J. Mathar, Sep 12 2008