A266479 Number of n-vertex simple graphs G_n for which n does not divide the number of labeled copies of G_n.

0, 2, 2, 6, 3, 20, 4

Offset: 1

Author

John P. McSorley, Dec 29 2015

Keywords

• nonn
• hard
• more

Comments

Let G_n be an n-vertex simple graph, with a(G_n) automorphisms. Then l(G_n) = n!/a(G_n) is the number of labeled copies of G_n. So a(n) is the number of G_n for which n does not divide l(G_n).

For prime p, a(p) is the number of circulants of order p.

The number of circulants of order n is A049287(n).

Examples

			If n=3 then both G_3 = K_3 and its complement have a(G_3) = 6, so l(G_3) = 3!/6 = 1, and so 3 does not divide l(G_3); no other graphs G_3 satisfy this, so a(3)=2.
		

References

• John P. McSorley, Smallest labelled class (and largest automorphism group) of a tree T_{s,t} and good labellings of a graph, preprint, (2016).
• R. C. Read, R. J. Wilson, An Atlas of Graphs, Oxford Science Publications, Oxford University Press, (1998).
• James Turner, Point-symmetric graphs with a prime number of points, Journal of Combinatorial Theory, vol. 3 (1967), 136-145.

Crossrefs

A000088 minus A266478.

Cf. A049287.