Particularly, I am dealing with Erd?s–Rényi random ??(??,??), so the expected Laplacian matrix of ??(??,??) is ??(?????????), where ???? and ???? are one and identity matrices, respectively.
In addition,if the distribution (unsure, but might be power law) of the eigenvalues of Laplacian matrix of the graph ??(??,??) is known, then it seems to me that expected value of the eigenvalues has some closed form formula depending on ?? and ?? in the asymptotic case.
See the papers by Erdos (no relation) and collaborators, e.g.:
Erd?s, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun, Spectral statistics of Erd?s-Rényi graphs. I: Local semicircle law, Ann. Probab. 41, No. 3B, 2279-2375 (2013). ZBL1272.05111.
Erd?s, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun, Spectral statistics of Erd?s-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues, Commun. Math. Phys. 314, No. 3, 587-640 (2012). ZBL1251.05162.
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