Combinatorica

, Volume 11, Issue 4, pp 369–382 | Cite as

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

  • Brendan D. McKay
  • Nicholas C. Wormald
Article

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 C30 05 C80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B. D. McKay: Asymptotics for symmetric 0–1 matrices with prescribed row sums,Ars Combinatoria,19A (1985) 15–25.Google Scholar
  2. [2]
    B. D. McKay, andN. C. Wormald: Automorphisms of random graphs with specified degrees,Combinatorica,4 (1984) 325–338.Google Scholar
  3. [3]
    B. D. McKay, andN. C. Wormald: Asymptotic enumeration by degree sequence of graphs of high degree,European J. Combinatorics,11 (1990) 565–580.Google Scholar
  4. [4]
    B. D. McKay, andN. C. Wormald: Uniform generation of random regular graphs of moderate degree,J. Algorithms,11 (1990) 52–67.Google Scholar

Copyright information

© Akadémiai Kiadó 1991

Authors and Affiliations

  • Brendan D. McKay
    • 1
  • Nicholas C. Wormald
    • 2
  1. 1.Computer Science DepartmentAustralian National UniversityAustralia
  2. 2.Department of MathematicsUniversity of MelbourneParkvilleAustralia

Personalised recommendations