The rank of diluted random graphs.(English)Zbl 1298.05283

Summary: We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs $$(G_n)_{n\geq 0}$$ 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 $$\varphi _{\ast }$$ 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 $$\varphi _{\ast }$$ for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of $$(G_n)_{n\geq 0}$$. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.

MSC:

 05C80 Random graphs (graph-theoretic aspects) 15B52 Random matrices (algebraic aspects) 47A10 Spectrum, resolvent
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References:

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