Counting connected graphs asymptotically

We find the asymptotic number of connected graphs with k vertices and k - 1 + l edges when k, l approach infinity, reproving a result of Bender, Canfield and McKay. We use the probabilistic method, analyzing breadth-first search on the random graph G(k, p) for an appropriate edge probability p. Central is analysis of a random walk with fixed beginning and end which is tilted to the left. 1 The Main Results In this paper, we investigate the number of graphs with a given complexity. Here, the complexity of a graph is its number of edges minus its number of vertices plus one. For k, l 0, we write C(k, l) for the number of labeled connected graphs with k vertices and complexity l. The study of C(k, l) has a long history. Cayley's Theorem gives the exact formula for the number of trees, C(k, 0) = kk-2. The asymptotic formula for the number of unicyclic graphs, C(k, 1), has been given by R

• Type: Journal
• Year: 2006
• Where: EJC
• Authors: Remco van der Hofstad, Joel Spencer
• Keywords: Appropriate Edge Probability, Asymptotic Number, EJC 2006, EJC 2007, Random Graph