Enumeration of labeled connected bicyclic and tricyclic graphs without bridges

Abstract

We consider undirected simple connected graphs. A bridge in a connected graph is an edge whose deletion makes the graph disconnected. A smooth graph is a graph without endpoints. Obviously, the set of graphs without bridges is a subset of the set of smooth graphs. Smooth graphs were enumerated by Wright [1]. Hanlon and Robinson [2] enumerated both labeled and unlabeled graphs without bridges. They obtained a functional equation and a nonlinear partial differential equation for the generating function of labeled graphs. However, their formulas are awkward and have not been reduced to a form convenient for computations. Apparently, that is why Sloane’s famous “On-Line Encyclopedia of Integer Sequences” [3] only provides data on the number of unlabeled graphs without bridges. The aim of the present paper is to obtain closed-form expressions for the number of labeled biand tricyclic graphs without bridges as well as the corresponding asymptotics for the number of such graphs as the number of vertices tends to infinity. Recall that a cutpoint of a connected graph is a vertex whose deletion (the edges incident to the vertex are deleted as well) makes the graph disconnected. A block is a connected graph without cutpoints or a maximal nontrivial connected subgraph without cutpoints [4, p. 200]. A graph is said to be uni-, bi-, or tricyclic if its cyclomatic number is 1, 2, or 3, respectively. For k ≥ 0, we denote the number of labeled blocks with n vertices and n+ k edges, the number of labeled connected graphs without bridges with n vertices and n+ k edges, and the number of labeled smooth graphs with n vertices and n+ k edges by u(n, n+ k), l(n, n+ k), and v(n, n + k), respectively. The cyclomatic number of any of these graphs is k + 1. Let Uk(w) and Lk(w) be the exponential generating functions,