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

Published online by Cambridge University Press:  06 September 2002

BRENDAN D. McKAY ,
IAN M. WANLESS  and
NICHOLAS C. WORMALD
BRENDAN D. McKAY
Affiliation:
Department of Computer Science, Australian National University, Canberra, ACT 0200, Australia (e-mail: bdm@cs.anu.edu.au)
IAN M. WANLESS
Affiliation:
Christ Church, Oxford OX1 1DP, England (e-mail: wanless@maths.ox.ac.uk)
NICHOLAS C. WORMALD
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Parkville, Vic 3052, Australia (e-mail: nick@ms.unimelb.edu.au)
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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 [ges ] −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.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 11 , Issue 4 , July 2002 , pp. 373 - 392
Copyright
2002 Cambridge University Press

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