We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the Erdös-Rényi random graph G(n, pn) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, pn (0, 1) and npn → ∞, we prove that the empirical distribution of the eigenvalues of the Laplacian matrix converges to a deterministic distribution, which is the free convolution of the semi-circle law and N(0, 1). However, for its normalized version, we prove that the empirical distribution converges to the semi-circle law.
Bibliographical noteFunding Information:
The author thanks Dr. Xue Ding for very helpful discussions on the Theorem 1.1. This research was supported in part by NSF #DMS-0449365.
© 2012 World Scientific Publishing Company.
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- Laplacian matrix
- Random matrix
- dilute graph
- free convolution
- normalized Laplacian matrix
- random graph
- semi-circle law
- spectral distribution