Values
([],1) => 0
([],2) => 0
([(0,1)],2) => 1
([],3) => 0
([(1,2)],3) => 1
([(0,2),(1,2)],3) => 0
([(0,1),(0,2),(1,2)],3) => 3
([],4) => 0
([(2,3)],4) => 2
([(1,3),(2,3)],4) => 0
([(0,3),(1,3),(2,3)],4) => 0
([(0,3),(1,2)],4) => 4
([(0,3),(1,2),(2,3)],4) => 1
([(1,2),(1,3),(2,3)],4) => 3
([(0,3),(1,2),(1,3),(2,3)],4) => 1
([(0,2),(0,3),(1,2),(1,3)],4) => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 12
([],5) => 0
([(3,4)],5) => 6
([(2,4),(3,4)],5) => 0
([(1,4),(2,4),(3,4)],5) => 0
([(0,4),(1,4),(2,4),(3,4)],5) => 0
([(1,4),(2,3)],5) => 4
([(1,4),(2,3),(3,4)],5) => 1
([(0,1),(2,4),(3,4)],5) => 2
([(2,3),(2,4),(3,4)],5) => 6
([(0,4),(1,4),(2,3),(3,4)],5) => 0
([(1,4),(2,3),(2,4),(3,4)],5) => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(1,3),(1,4),(2,3),(2,4)],5) => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(0,4),(1,3),(2,3),(2,4)],5) => 0
([(0,1),(2,3),(2,4),(3,4)],5) => 6
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 60
([],6) => 0
([(4,5)],6) => 24
([(3,5),(4,5)],6) => 0
([(2,5),(3,5),(4,5)],6) => 0
([(1,5),(2,5),(3,5),(4,5)],6) => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
([(2,5),(3,4)],6) => 8
([(2,5),(3,4),(4,5)],6) => 2
([(1,2),(3,5),(4,5)],6) => 2
([(3,4),(3,5),(4,5)],6) => 18
([(1,5),(2,5),(3,4),(4,5)],6) => 0
([(0,1),(2,5),(3,5),(4,5)],6) => 6
([(2,5),(3,4),(3,5),(4,5)],6) => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(2,4),(2,5),(3,4),(3,5)],6) => 0
([(0,5),(1,5),(2,4),(3,4)],6) => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24
([(0,5),(1,4),(2,3)],6) => 24
([(1,5),(2,4),(3,4),(3,5)],6) => 0
([(0,1),(2,5),(3,4),(4,5)],6) => 2
([(1,2),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 4
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 4
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 5
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 1
>>> Load all 208 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of odd automorphisms of a graph.
Let $D$ be an arbitrary orientation of a graph $G$. Then an automorphism of $G$ is odd, if it reverses the orientation of an odd number of edges of $D$.
The graphs on $n$ vertices without any odd automorphisms are equinumerous with the number of non-isomorphic $n$-team tournaments, see [2].
The odd automorphisms of the complete graphs are precisely the even permutations.
Let $D$ be an arbitrary orientation of a graph $G$. Then an automorphism of $G$ is odd, if it reverses the orientation of an odd number of edges of $D$.
The graphs on $n$ vertices without any odd automorphisms are equinumerous with the number of non-isomorphic $n$-team tournaments, see [2].
The odd automorphisms of the complete graphs are precisely the even permutations.
References
[1] Number of outcomes of unlabeled n-team round-robin tournaments. OEIS:A000568
[2] Royle, G. F., Praeger, C. E., Glasby, S. P., Freedman, S. D., Devillers, A. Tournaments and Even Graphs are Equinumerous arXiv:2204.01947
[2] Royle, G. F., Praeger, C. E., Glasby, S. P., Freedman, S. D., Devillers, A. Tournaments and Even Graphs are Equinumerous arXiv:2204.01947
Code
def is_odd_automorphism(D, g):
count = 0
for a, b in D.edges(labels=False):
c, d = g(a), g(b)
if D.has_edge(c, d) and not D.has_edge(d, c):
pass
elif D.has_edge(d, c) and not D.has_edge(c, d):
count += 1
else:
raise ValueError("%s is mapped to %s" % ((a,b), (c,d)))
return is_odd(count)
def statistic(G):
D = next(G.orientations())
count = 0
for g in G.automorphism_group():
if is_odd_automorphism(D, g):
count += 1
return count
Created
Apr 06, 2022 at 10:26 by Martin Rubey
Updated
Apr 27, 2022 at 10:37 by Martin Rubey
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!