Size of nodal domains of the eigenvectors of a graph

Consider an eigenvector of the adjacency matrix of a G ( n , p ) graph. A nodal domain is a connected component of the set of vertices where this eigenvector has a constant sign. It is known that with high probability, there are exactly two nodal domains for each eigenvector corresponding to a nonleading eigenvalue. We prove that with high probability, the sizes of these nodal domains are approximately equal to each other.