Asymptotic Enumeration of Graphs with a Given Upper Bound on the Maximum Degree

Authors:
Brendan D. McKay;Ian M. Wanless;Nicholas C. Wormald
Affiliations:
Department of Computer Science, Australian National University, Canberra, ACT 0200, Australia (e-mail: bdm@cs.anu.edu.au);Christ Church, Oxford OX1 1DP, England (e-mail: wanless@maths.ox.ac.uk);Department of Mathematics and Statistics, University of Melbourne, Parkville, Vic 3052, Australia (e-mail: nick@ms.unimelb.edu.au)
Venue:
Combinatorics, Probability and Computing
Year:
2002

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Abstract

Consider the class of graphs on n vertices which have maximum degree at most 1/2n−1+τ, where τ ≥ −n1/2+ε for sufficiently small ε 0. We find an asymptotic formula for the number of such graphs and show that their number of edges has a normal distribution whose parameters we determine. We also show that expectations of random variables on the degree sequences of such graphs can often be estimated using a model based on truncated binomial distributions.