ABSTRACT
This chapter discusses the easiest enumeration problems, those for labeled graphs. It presents G. Polya's classical enumeration theorem and utilizes it to derive counting series for trees and various other kinds of graphs. Polya's theorem has been generalized to the Power Group Enumeration Theorem which is useful for certain counting problems where the equivalence classes are determined by two permutation groups. Many enumeration problems are formulated in such a way that the answer can be given by finding a formula for the number of orbits determined by a permutation group. Polya's theorem in turn depends on a generalization of a well-known counting formula due to W. Burnside. The classical counting problems to which Polya's Theorem applies all have the same general form. In applications of Polya's Enumeration Theorem, certain permutation groups occur frequently. The chapter concludes with lists of both solved and unsolved problems in graphical enumeration.
