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Subgraph counts for dense random graphs with specified degrees

Part of: Graph theory Combinatorics

Published online by Cambridge University Press:  05 November 2020

Catherine Greenhill ,
Mikhail Isaev  and
Brendan D. McKay
Catherine Greenhill*
Affiliation:
School of Mathematics and Statistics, UNSW Sydney, NSW2052, Australia
Mikhail Isaev
Affiliation:
School of Mathematical Sciences, Monash University, VIC3800, Australia
Brendan D. McKay
Affiliation:
Research School of Computer Science, Australian National University, ACT2601, Australia
*
*Corresponding author. Email: c.greenhill@unsw.edu.au
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Abstract

We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d1, …, dn) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes dj= λn + O(n1/2+ε) for some λ bounded away from 0 and 1.

MSC classification

Primary: 05A16: Asymptotic enumeration
Secondary: 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 3 , May 2021 , pp. 460 - 497
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Research supported by Australian Research Council Discovery Project DP190100977.

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