A182012 Number of graphs on 2n unlabeled nodes all having odd degree.

1, 3, 16, 243, 33120, 87723296, 3633057074584, 1967881448329407496, 13670271807937483065795200, 1232069666043220685614640133362240, 1464616584892951614637834432454928487321792, 23331378450474334173960358458324497256118170821672192, 5051222500253499871627935174024445724071241027782979567759187712

Offset: 1

Author

Georgi Guninski, Apr 06 2012

Keywords

• nonn
• easy

Comments

As usual, "graph" means "simple graph, without self-loops or multiple edges".

The graphs on 2n vertices all having odd degrees are just the complements of those having all even degrees. That's why the property of all odd degrees is seldom mentioned. Therefore this sequence is just every second term of A002854. - Brendan McKay, Apr 08 2012

Examples

			The 3 graphs on 4 vertices are [(0, 3), (1, 3), (2, 3)], [(0, 2), (1, 3)], [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]: the tree with root of order 3, the disconnected graph consisting of two complete 2-vertex graphs, and the complete graph.
		

Links

• Sequence Fans Mailing List, Discussion, April 2012.
• N. J. A. Sloane, The 16 graphs on 6 nodes

Crossrefs

Cf. A210345, A210346, A000088. Bisection of A002854.

Programs

Sage

def graphsodddegree(MAXN=5):
    """
    requires optional package "nauty"
    """
    an=[]
    for n in range(1,MAXN+1):
        gn=graphs.nauty_geng("%s"%(2*n))
        cac={}
        a=0
        for G in gn:
            d = G.degree_sequence()
            if all(i % 2 for i in d):
                a += 1
        print('a(%s)=%s'%(n,a))
        an += [a]
    return an

Formula

a(n) = A002854(2n).

Extensions

Terms from a(6) on added from A002854. - N. J. A. Sloane, Apr 08 2012