In this work we study statistical properties of the extreme eigenstates of the randomly weighted adjacency matrices of random graphs. We focus on two random graph models: Erdős-Rényi graphs and random geometric graphs. Indeed, the adjacency matrices of both graph models are diluted versions of the Gaussian orthogonal ensemble (GOE) of random-matrix theory such that a transition from the Poisson ensemble (PE) to the GOE is observed by increasing the graph average degree $\langle k\rangle$. First, we write expressions for the spectral density in terms of $\langle k\rangle$ for the regimes below and above the percolation threshold. Then we show that the distributions of the largest ${\lambda }_1$ and second-largest ${\lambda }_2$ eigenvalues approach the Tracy-Widom distribution of type 1 for $\langle k\rangle \gg 1$, while $\langle {\lambda }_1\rangle =\sqrt{2\langle k\rangle }$. Additionally, we demonstrate that the distributions of the normalized distance between ${\lambda }_1$ and ${\lambda }_2$, the distribution of the ratio between higher consecutive eigenvalue spacings, and the distributions of the inverse participation ratios of the extreme eigenstates display a clear PE-to-GOE transition as a function of $\langle k\rangle$. We conclude that any of these distributions can be effectively used to probe the delocalization transition of the graph models without the need of the full spectrum.

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• Erdos-Renyi graph
• Random graphs
• Stochastic networks
• Random matrix theory