For $N_\lambda$ denoting the number of eigenvalues less than $\lambda$, Weyl's law gives the asymptotics of $N_\lambda$ as $\lambda$ tends to infinity. The usual approach to establish this asymptotics in the continuum divides space into boxes. This fails for a graph, since cutting the graph is not a small perturbation.   
An analogue of Weyl asymptotics for graph Laplacians that satisfy a strong isoperimetric inequality is given in <A HREF="https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-GraphTh/Keller/KellerLenzWojciechowski_GraphsAndDiscreteDirichletSpaces_wu_version.pdf">Graphs and Discrete Dirichlet Spaces</A> by Keller, Lenz, and Wojciechowski, page 439. See also <A HREF="https://hal.science/hal-01010730/document">Essential spectrum and Weyl asymptotics for discrete Laplacians</A> by Bonnefont and Golenia.

The above refers to the eigenvalues. For the eigenvectors, the localisation phenomenon has been studied in <A HREF="https://www.nature.com/articles/s41598-017-01010-0">Localization of Laplacian eigenvectors on random networks</A>, but there is no "rigorous" theory.

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Concerning the connection with random matrix theory: Randomly weighted graphs [fixed number of neighbors $d$ of each vertex, with weights defined on the edges that are uniformly and independently drawn from (-1,1)] have a mean spectral density $\rho(\mu)$ of the graph Laplacian given by the Kesten-McCay law
$$\rho(\mu)=\frac{d}{2\pi}(d^2-\mu^2)^{-1}\sqrt{4(d-1)-\mu^2},\;\;\text{for}\;\;|\mu|<2\sqrt{d-1}.$$
See for example <A HREF="https://arxiv.org/abs/1609.09052">Local Kesten-McKay law for random regular graphs</A>. In this case the eigenvalues are delocalised. The Wigner semi-circle law of Gaussian random matrix ensembles is obtained in the limit $d\rightarrow\infty$.