Use Cayley's Tree-Counting Theorem to count the following types of graphs on labeled vertex set V = {v1,...,vn}:
Two-component forests: that is, two connected components, each a tree. For example, for n = 3, we have the 3 graphs:
1−−2 3 , 1−−3 2 , 2−−3 1
Unicyclic graphs: that is, connected with exactly one cycle. For n = 3, we have the unique graph C3. For n = 4, we have triangles plus an extra edge (4×3 graphs); or C4 (4!/8 graphs), for 15 graphs in all. This problem is hard, so say as much as you can, giving examples.

Question

Use Cayley's Tree-Counting Theorem to count the following types of graphs on labeled vertex set V = {v1,...,vn}:
Two-component forests: that is, two connected components, each a tree. For example, for n = 3, we have the 3 graphs:
1−−2   3   ,   1−−3   2   ,   2−−3   1
Unicyclic graphs: that is, connected with exactly one cycle. For n = 3, we have the unique graph C3. For n = 4, we have triangles plus an extra edge (4×3 graphs); or C4 (4!/8 graphs), for 15 graphs in all. This problem is hard, so say as much as you can, giving examples.