Graph enumeration

In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected or directed graphs of certain types, typically as a function of the number of vertices of the graph. These problems may be solved either exactly (as an algebraic enumeration problem) or asymptotically. The pioneers in this area of mathematics were Pólya,Cayley and Redfield.

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. In general, labeled problems tend to be easier to solve than unlabeled problems. As with combinatorial enumeration more generally, the Pólya enumeration theorem is an important tool for dealing with symmetries such as this.

Exact enumeration formulas

Some important results in this area include the following.