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Expected Maximum Block Size in Critical Random Graphs

Published online by Cambridge University Press:  25 July 2019

V. Rasendrahasina
Affiliation:
ENS – Université d’ Antananarivo, 101 Antananarivo, Madagascar
A. Rasoanaivo
Affiliation:
LIMA – Université d’ Antananarivo, 101 Antananarivo, Madagascar
V. Ravelomanana
Affiliation:
IRIF UMR CNRS 8243 – Universite Denis Diderot, 75013 Paris, France
Corresponding
E-mail address:

Abstract

Let G(n,M) be a uniform random graph with n vertices and M edges. Let ${\wp_{n,m}}$ be the maximum block size of G(n,M), that is, the maximum size of its maximal 2-connected induced subgraphs. We determine the expectation of ${\wp_{n,m}}$ near the critical point M = n/2. When n − 2Mn2/3, we find a constant c1 such that

$$c_1 = \lim_{n \rightarrow \infty} \left({1 - \frac{2M}{n}} \right) \,\E({\wp_{n,m}}).$$
Inside the window of transition of G(n,M) with M = (n/2)(1 + λn−1/3), where λ is any real number, we find an exact analytic expression for
$$c_2(\lambda) = \lim_{n \rightarrow \infty} \frac{\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}}{n^{1/3}}.$$
This study relies on the symbolic method and analytic tools from generating function theory, which enable us to describe the evolution of $n^{-1/3}\,\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}$ as a function of λ.

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© Cambridge University Press 2019 

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