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A141580
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Number of unlabeled non-mating graphs with n vertices.
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7
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0, 1, 2, 6, 18, 78, 456, 4299, 68754, 1990286, 106088988, 10454883132, 1904236651216, 641859005526860, 401547534010157680, 467956331904669136874, 1019785644052109276678788, 4171197546082606538129623140
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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.
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)
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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