# Nodal domains of eigenvectors of Erdos-Renyi random graphs

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A nodal domain of a function is a connected component of the set where this function has a constant sign. Nodal domains of eignefunctions of the Laplacian are classical objects in geometry of manifolds. Their number of nodal domains corresponding to the k-th smallest eigenvalue is bounded by k and it typically increases as k gets larger. About 10 years ago, Dekel, Lee, and Linial proved that with high probability, the number of nodal domains of Erdos-Renyi graphs remains bounded as the size of the graph and the eigenvalues increase. This runs contrary to the intuition we draw from the world of manifolds. We will survey some recent results on the structure of nodal domains of such graphs. Based in part on the joint work with Han Huang.