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

1,2

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

Links

Table of n, a(n) for n=1..13.

Sequence Fans Mailing List, Discussion, April 2012.

N. J. A. Sloane, The 16 graphs on 6 nodes

Formula

a(n) = A002854(2n).

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A361366 A000273 A071897 * A272385 A013923 A053466

Adjacent sequences: A182009 A182010 A182011 * A182013 A182014 A182015

Keyword

nonn,easy

Author

Georgi Guninski, Apr 06 2012

Extensions

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

Status

approved