|
|
A182012
|
|
Number of graphs on 2n unlabeled nodes all having odd degree.
|
|
4
|
|
|
1, 3, 16, 243, 33120, 87723296, 3633057074584, 1967881448329407496, 13670271807937483065795200, 1232069666043220685614640133362240, 1464616584892951614637834432454928487321792, 23331378450474334173960358458324497256118170821672192, 5051222500253499871627935174024445724071241027782979567759187712
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
Sequence Fans Mailing List, Discussion, April 2012.
|
|
FORMULA
|
|
|
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
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|