Counting Connected Graphs Asymptotically
Abstract
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.
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