The rank of diluted random graphs

Authors:
Charles Bordenave;Marc Lelarge
Affiliations:
Institut de Mathématiques - Université de Toulouse & CNRS;INRIA-École Normale Supérieure
Venue:
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Year:
2010

Citing 3
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Maximum matching in sparse random graphs

SFCS '81 Proceedings of the 22nd Annual Symposium on Foundations of Computer Science
The rank of random graphs

Random Structures & Algorithms
Resolvent of large random graphs

Random Structures & Algorithms

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Abstract

We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs converging locally to a tree, we give new formulas for the asymptotic of the multiplicity of the eigenvalue 0. In particular, the result depends only on the limiting tree structure, showing that the normalized rank is 'continuous at infinity'. Our work also gives a new formula for the mass at zero of the spectral measure of a Galton-Watson tree. Our techniques of proofs borrow ideas from analysis of algorithms, random matrix theory, statistical physics and analysis of Schrödinger operators on trees.