Enumeration of generalized graphs

Abstract

In this paper we are concerned with a problem in the Pólya theory of enumeration. Our main result is the determination of the cycle index of a group S H which acts on a set of equivalence classes V N / H. (Let V and N be finite, non-empty sets and let S and H be permutation groups on V and N respectively; V N is the set of injective mappings of N into V; elements f, g ∈ V N are H-equivalent if there exists θ ∈ H such that fθ = g, and V N / H is the set of H-equivalence classes. Any σ ∈ S induces, in a natural way, a permutation τ H ( σ) of V N / H, and S H is the group of all τ H ( σ)'s.) This result is used to enumerate classes of structures such as linear graphs, directed graphs, and generalized graphs. A typical consequence of our results is the determination of the cycle structure of the permutation of the subsets of a finite set induced by a permutation of the set.