TITLE: limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?
QUESTION [8 upvotes]: Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where $A$ is the adjacency matrix of the graph and $D$ is diagonal with $D_{ii} = $ degree of node $i$). Let finally $0 = \lambda_0 \leq \lambda_1 \leq \lambda_2 \leq \cdots$ be the eigenvalues of $L$. Is an analytic expression known for the limiting empirical distribution of these eigenvalues, i.e. for
$\lim_{n \to \infty} \frac{1}{n} \#\{ 1 \leq j \leq n : \lambda_j \leq t \}, t \in \mathbb{R}?$

REPLY [3 votes]: As far as I know, the spectra of Laplacian of a graph follows a distribution that is the free additive convolution of a Gaussian and the semicircle law.
Take a look, for example at the article:
"Spectra of random graphs with arbitrary expected degrees"
Raj Rao Nadakuditi, M. E. J. Newman
https://arxiv.org/abs/1208.1275
In your case the arbitrary expected degree is constant, therefore the computation should result simpler than in the more general case.