The rank of diluted random graphs
Charles Bordenave, Marc Lelarge, and Justin Salez
Source: Ann. Probab. Volume 39, Number 3
(2011), 1097-1121.
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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Keywords: Random graphs; adjacency matrix; random matrices; local weak convergence; Karp and Sipser algorithm
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1300281733
Digital Object Identifier: doi:10.1214/10-AOP567
Zentralblatt MATH identifier: 05885404
Mathematical Reviews number (MathSciNet): MR2789584
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