The Annals of Probability

The rank of diluted random graphs

Charles Bordenave, Marc Lelarge, and Justin Salez

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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.

Article information

Ann. Probab. Volume 39, Number 3 (2011), 1097-1121.

First available in Project Euclid: 16 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 15A52
Secondary: 47A10: Spectrum, resolvent

Random graphs adjacency matrix random matrices local weak convergence Karp and Sipser algorithm


Bordenave, Charles; Lelarge, Marc; Salez, Justin. The rank of diluted random graphs. Ann. Probab. 39 (2011), no. 3, 1097--1121. doi:10.1214/10-AOP567.

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