Graph enumeration

Labeled vs unlabeled problems

In some graphical enumeration problems, the vertices of the graph are considered to be labeled in such a way as to be distinguishable from each other, while in other problems any permutation of the vertices is considered to form the same graph, so the vertices are considered identical or unlabeled. In general, labeled problems tend to be easier. As with combinatorial enumeration more generally, the Pólya enumeration theorem is an important tool for reducing unlabeled problems to labeled ones: each unlabeled class is considered as a symmetry class of labeled objects.

The number of unlabelled graphs with ${\displaystyle n}$ vertices is still not known in a closed-form solution, but as almost all graphs are asymmetric this number is asymptotic to $${\displaystyle \frac{2^{(\genfrac{}{}{0ex}{}{n}{2})}}{n!}.}$$