Combinatorica

, Volume 11, Issue 4, pp 369–382

Asymptotic enumeration by degree sequence of graphs with degreeso(n1/2)

Authors

  • Brendan D. McKay
    • Computer Science DepartmentAustralian National University
  • Nicholas C. Wormald
    • Department of MathematicsUniversity of Melbourne
Article

DOI: 10.1007/BF01275671

Cite this article as:
McKay, B.D. & Wormald, N.C. Combinatorica (1991) 11: 369. doi:10.1007/BF01275671

Abstract

We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree iso(|E(G)|1/3). The previously best enumeration, by the first author, required maximum degreeo(|E(G)|1/4). In particular, ifk=o(n1/2), the number of regular graphs of degreek and ordern is asymptotically
$$\frac{{(nk)!}}{{(nk/2)!2^{nk/2} (k!)^n }}\exp \left( { - \frac{{k^2 - 1}}{4} - \frac{{k^3 }}{{12n}} + 0\left( {k^2 /n} \right)} \right).$$
Under slightly stronger conditions, we also determine the asymptotic number of unlabelled graphs with a given degree sequence. The method used is a switching argument recently used by us to uniformly generate random graphs with given degree sequences.

AMS subject classification (1991)

05 C3005 C80

Copyright information

© Akadémiai Kiadó 1991