# Asymptotic enumeration by degree sequence of graphs with degrees o(n1/2)

Brendan D. McKay, Nicholas C. Wormald

Research output: Contribution to journalArticleResearchpeer-review

109 Citations (Scopus)

### Abstract

We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree is 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(n1/2), the number of regular graphs of degree k and order n is asymptotically {Mathematical expression} 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.

Original language English 369-382 14 Combinatorica 11 4 https://doi.org/10.1007/BF01275671 Published - Dec 1991 Yes

### Keywords

• AMS subject classification (1991): 05C30, 05C80

### Cite this

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title = "Asymptotic enumeration by degree sequence of graphs with degrees o(n1/2)",
abstract = "We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree is 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(n1/2), the number of regular graphs of degree k and order n is asymptotically {Mathematical expression} 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.",
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In: Combinatorica, Vol. 11, No. 4, 12.1991, p. 369-382.

Research output: Contribution to journalArticleResearchpeer-review

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T1 - Asymptotic enumeration by degree sequence of graphs with degrees o(n1/2)

AU - McKay, Brendan D.

AU - Wormald, Nicholas C.

PY - 1991/12

Y1 - 1991/12

N2 - We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree is 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(n1/2), the number of regular graphs of degree k and order n is asymptotically {Mathematical expression} 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.

AB - We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree is 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(n1/2), the number of regular graphs of degree k and order n is asymptotically {Mathematical expression} 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.

KW - AMS subject classification (1991): 05C30, 05C80

UR - http://www.scopus.com/inward/record.url?scp=0000216866&partnerID=8YFLogxK

U2 - 10.1007/BF01275671

DO - 10.1007/BF01275671

M3 - Article

VL - 11

SP - 369

EP - 382

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 4

ER -