Spectral distributions of adjacency and Laplacian matrices of random graphs

  • Ding X
  • Jiang T
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Abstract

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.

Author-supplied keywords

  • Adjacency matrix
  • Free convolution
  • Laplacian matrix
  • Largest eigenvalue
  • Random graph
  • Random matrix
  • Semi-circle law
  • Spectral distribution

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Authors

  • Xue Ding

  • Tiefeng Jiang

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