# Asymptotic enumeration by degree sequence of graphs with degrees*o*(*n*^{1/2})

## Authors

Article

- Received:

DOI: 10.1007/BF01275671

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

- 53 Citations
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## Abstract

We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree is 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.

*o*(|*E(G)*|^{1/3}). The previously best enumeration, by the first author, required maximum degree*o*(|*E(G)*|^{1/4}). In particular, if*k*=*o*(*n*^{1/2}), the number of regular graphs of degree*k*and order*n*is asymptotically$$\frac{{(nk)!}}{{(nk/2)!2^{nk/2} (k!)^n }}\exp \left( { - \frac{{k^2 - 1}}{4} - \frac{{k^3 }}{{12n}} + 0\left( {k^2 /n} \right)} \right).$$

### AMS subject classification (1991)

05 C3005 C80## Copyright information

© Akadémiai Kiadó 1991