The Eigenvector centrality for a graph $G = (V,E)$ is defined as the eigenvector corresponding to the larges eigenvalue of the adjacency matrix $\mathbf{A}$.
The PageRank is a special case of eigenvector centrality of a graph $G$ and is defined as the eigenvector corresponding to the largest eigenvalue of the Google matrix $\mathbf{M}$ (usually found by power iteration), which is a row-wise normalized adjacency matrix combined with a "random jump assumption" $$ \mathbf{M} = \alpha(\mathbf{K^{-1}}\mathbf{A})^T + (1-\alpha)\frac{1}{N}\mathbf{E} $$ where $\mathbf{K}$ is the diagonal matrix with outdegrees on the diagonal and $\mathbf{E}$ the matrix of all ones.
I was wondering, if we have an undirected graph ($\mathbf{A} = \mathbf{A}^T$) and we set the random jump probability to 0 (i.e. $\alpha = 1$), will the page rank centrality be equal the the eigenvector centrality up to some scaling factor? Will both eigenvectors point in the same direction? How does the row-wise normalization affect the eigenvectors of the adjacency matrix $\mathbf{A}$?