SPECTRAL DISTRIBUTIONS OF ADJACENCY AND LAPLACIAN MATRICES OF RANDOM GRAPHS

By Xue Ding, Tiefeng Jiang. 2010

121 citations recorded by the source corpus.

In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that: (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval almost surely; (iii) the empirical distributions of the eigenvalues of the Laplacian matrices converge weakly to the free convolution of the standard Gaussian distribution and the Wigner’s semi-circular law; (iv) the empirical distributions of the eigenvalues of the adjacency matrices converge weakly to the Wigner’s semi-circular law.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cd67aaa9b8bc9ad53a2f0b6392e3ceb8f8c589f7

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In Geometry and Topology

1. T. Von
2. Debashis Paul
3. Alexander Aue

In Mathematical Physics

1. Debashis Paul
2. Alexander Aue
3. T. Von

In Statistics and Probability

1. Alexander Aue
2. Debashis Paul
3. T. Von
4. Lajos Horv'ath
5. A. Onatski
6. D. Xiu
7. Yacine Aït-Sahalia
8. M. Y. Mo

Citations in the corpus

• Random matrix theory in statistics: A review (2014, methodology): “Random matrices have also been used to describe properties of large random graphs, especially the behavior of eigenvalues of graph Laplacian and the adjacency matrix (Ding and Jiang, 2010; Jiang, 2012).”
• ALGEBRAIC CONNECTIVITY OF ERDÖS-RÉNYI GRAPHS NEAR THE CONNECTIVITY THRESHOLD (2014, background): “In a similar spirit, [17, 24, 38] establish a law 4”