I am studying the relation of the symmetric matrix and its eigenvalues and eigenvectors to understand the graph features from its adjacency matrix.
Suppose that matrix B with size n whose elements are all one, and A is a symmetric matrix with size n. It consists of zero and one, and its diagonal components are all zero. If A is completely divided by eigenspace with a dimension greater than one, B - A has at least one eigenvector with one dimension.
It seems correct, but I cannot construct the proof.