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May 2011 The rank of diluted random graphs
Charles Bordenave, Marc Lelarge, Justin Salez
Ann. Probab. 39(3): 1097-1121 (May 2011). DOI: 10.1214/10-AOP567

Abstract

We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs (Gn)n≥0 converging locally to a Galton–Watson tree T (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function φ of T. In the first part, we show that the adjacency operator associated with T is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on φ for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of (Gn)n≥0. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.

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Charles Bordenave. Marc Lelarge. Justin Salez. "The rank of diluted random graphs." Ann. Probab. 39 (3) 1097 - 1121, May 2011. https://doi.org/10.1214/10-AOP567

Information

Published: May 2011
First available in Project Euclid: 16 March 2011

zbMATH: 1298.05283
MathSciNet: MR2789584
Digital Object Identifier: 10.1214/10-AOP567

Subjects:
Primary: 05C80 , 15A52
Secondary: 47A10

Keywords: ‎adjacency matrix , Karp and Sipser algorithm , Local weak convergence , Random graphs , random matrices

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 3 • May 2011
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