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Subgraph counts for dense random graphs with specified degrees

Published online by Cambridge University Press:  05 November 2020

Catherine Greenhill
Affiliation:
School of Mathematics and Statistics, UNSW Sydney, NSW 2052, Australia
Mikhail Isaev
Affiliation:
School of Mathematical Sciences, Monash University, VIC 3800, Australia
Brendan D. McKay
Affiliation:
Research School of Computer Science, Australian National University, ACT 2601, Australia
Corresponding
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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(d 1 , …, d n ) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes d j = λn + O(n 1/2+ε ) for some λ bounded away from 0 and 1.

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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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