# 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. The pioneers in this area of mathematics were Pólya, Cayley  and Redfield.

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.

Some important results in this area include the following.

• The number of labeled n-vertex undirected graphs is 2n(n − 1)/2.
• The number of labeled n-vertex directed graphs is 2n(n − 1).
• The number Cn of connected labeled n-vertex undirected graphs satisfies the recurrence relation
$C_{n}=2^{n \choose 2}-{\frac {1}{n}}\sum _{k=1}^{n-1}k{n \choose k}2^{n-k \choose 2}C_{k}.$ from which one may easily calculate, for n = 1, 2, 3, ..., that the values for Cn are
1, 1, 4, 38, 728, 26704, 1866256, ...(sequence A001187 in OEIS)
$2^{n-4}+2^{\lfloor (n-4)/2\rfloor }.$ ## Enumerating Graphs on A Degree Sequence

Given a degree sequence d one can enumerate all the graphs with degree sequence d using an algorithm. One such algorithm given by Blitzstein and Diaconis  is as follows;

INPUT: a graphical sequence d=(d1d2, ..., dn).

1. Let E be an empty list of edges.
2. If d = 0, terminate with output E.
3. Choose the least i with di a minimal positive entry.
4. Compute the candidate list J={j≠i : {i,j} ∉ E and (d1d2, ...,di − 1,..., dj − 1, ..., dn)  is graphical.}
5. Pick j ∈ J with probability proportional to its degree in d.
6. Add edge {i,j} to E and update d to (d1d2, ...,di − 1,..., dj − 1, ..., dn
7. Repeat steps 4-6 until the degree of i is 0.