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On the Laplacian Eigenvalues of Gn,p

Published online by Cambridge University Press:  01 November 2007

AMIN COJA-OGHLAN
AMIN COJA-OGHLAN*
Affiliation:
Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA 15213, USA (e-mail: coja@informatik.hu-berlin.de)
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Abstract

We investigate the Laplacian eigenvalues of sparse random graphs Gnp. We show that in the case that the expected degree d = (n-1)p is bounded, the spectral gap of the normalized Laplacian is o(1). Nonetheless, w.h.p. G = Gnp has a large subgraph core(G) such that the spectral gap of is as large as 1-O (d−1/2). We derive similar results regarding the spectrum of the combinatorial Laplacian L(Gnp). The present paper complements the work of Chung, Lu and Vu [8] on the Laplacian spectra of random graphs with given expected degree sequences. Applied to Gnp, their results imply that in the ‘dense’ case d ≥ ln2n the spectral gap of is 1-O (d−1/2) w.h.p.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 6 , November 2007 , pp. 923 - 946
Copyright
Copyright © Cambridge University Press 2007

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