Counting connected graphs asymptotically

Authors:
Remco van der Hofstad;Joel Spencer
Affiliations:
Department of Mathematics and Computer Science, Eindhoven University of Technology, MB Eindhoven, The Netherlands;Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, New York, NY
Venue:
European Journal of Combinatorics - Special issue on extremal and probabilistic combinatorics
Year:
2006

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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, re-proving 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 the analysis of a random walk with fixed beginning and end which is tilted to the left.