In this chapter we consider enumerative problems of graph theory, that is, problems arising when counting graphs with specific properties. We also consider similar problems about mappings of finite sets with various constraints. Particular attention is given to mappings of bounded height h, whose cycle lengths are the elements of a given sequence A. For h = 0 these mappings become substitutions whose cycle lengths belong to a given sequence A.

The method of generating functions can be effectively applied to these problems. As a result, we obtain either explicit formulae or some expressions for generating functions which allow us to find the asymptotic expressions of the corresponding coefficients, for instance by using the saddle point method. The results of the application of the saddle point method to the derivation of such asymptotic expressions are not given here. They can be found, for example, in papers (Sachkov, 1972; Sachkov, 1971b; Sachkov, 1971a).

The generating functions for graphs

Basic definitions

Let us formulate the basic definitions about the graphs; we follow the most common terminology of graph theory (Berge, 1958; Ore, 1962; Harary, 1969). A graph Γ = Γ(X, W) consists of a set X containing n ≥ 1 elements called vertices and of a set W of unordered pairs of vertices called edges. Usually a graph is geometrically represented on the plane by points corresponding to the vertices, and lines corresponding to the edges which join pairs of vertices from W; the intersections of the lines at points that differ from the vertices are not taken into account.