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A079565
Number of unlabeled and connected graphs on n vertices which are either bipartite or co-bipartite.
1
1, 1, 2, 6, 16, 49, 129, 481, 1845, 9506, 57896, 463909, 4769436, 65179170, 1187099045, 29082860878, 960963147303, 42920936851975, 2594399793419459, 212465886865393053, 23596018831885668391, 3557502387712889568013, 728850489548729072323085
OFFSET
1,3
COMMENTS
G is bipartite iff the vertices can be partitioned into two sets such that all the edges in the graph go from one of these sets to the other. G is cobipartite iff the complement of G is bipartite.
For n >= 5, no graph can be both bipartite and co-bipartite. - Falk Hüffner, Jan 22 2016
LINKS
FORMULA
For n >= 5, a(n) = A079571(n) + A005142(n). - Falk Hüffner, Jan 22 2016
EXAMPLE
Let G be a graph with 5 vertices, 4 of which form a path and the 5th adjacent only to the two vertices in the middle of the path. Then G is not bipartite nor cobipartite because there is a triangle in both G and its complement.
MATHEMATICA
A005142 = Import["https://oeis.org/A005142/b005142.txt", "Table"][[All, 2]];
A033995 = Import["https://oeis.org/A033995/b033995.txt", "Table"][[All, 2]];
a[n_] := If[n<5, {1, 1, 2, 6}[[n]], A005142[[n+1]] + A033995[[n+1]] - Floor[n/2]];
a /@ Range[1, 50] (* Jean-François Alcover, Sep 17 2019 *)
CROSSREFS
Sequence in context: A272411 A151528 A132803 * A052890 A052814 A192401
KEYWORD
nonn
AUTHOR
Jim Nastos, Jan 24 2003
EXTENSIONS
More terms using formula by Falk Hüffner, Jan 22 2016
Terms a(21) and beyond from Andrew Howroyd, Sep 05 2018
STATUS
approved