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Unicyclic components in random graphs

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Published 20 April 2004 • 2004 IOP Publishing Ltd
, , Citation E Ben-Naim and P L Krapivsky 2004 J. Phys. A: Math. Gen. 37 L189 DOI 10.1088/0305-4470/37/18/L01

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Author affiliations

1 Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

2 Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA

Dates

  1. Received 18 March 2004
  2. Published

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0305-4470/37/18/L189

Abstract

The distribution of unicyclic components in a random graph is obtained analytically. The number of unicyclic components of a given size approaches a self-similar form in the vicinity of the gelation transition. At the gelation point, this distribution decays algebraically, Uk ≃ (4k)−1 for k ≫ 1. As a result, the total number of unicyclic components grows logarithmically with the system size.

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10.1088/0305-4470/37/18/L01