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Counting Connected Graphs Asymptotically

Remco van der Hofstad·Joel Spencer
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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 {\em 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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Remco van der Hofstad, Joel Spencer. 2005-02-28. Counting Connected Graphs Asymptotically. https://arxiv.org/abs/math/0502579

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