I have a matrix $G$ that is a row-normalized adjacency matrix of a connected graph. This means that all entries are $\geq 0$ and every row sums to one. The diagonal only contains zeros.
From the Perron-Frobenius theorem, I know that $G$ has a unique (largest) eigenvalue 1 (the eigenvector is the vector of ones). Therefore, I know that the Laplacian matrix $(I-G)$ has a unique eigenvalue 0 (the eigenvector is again the vector of ones), so the null space of $(I-G)$ is of dimension 1 and is proportional to this vector of ones.
Now my question is how to prove that the null space of $(I-G)^2$ is also of dimension 1 (and not of 2). Any suggestions how to proceed?