Let $\lambda_m$ denote the $m$-th eigenvalue of the adjacency matrix $A$ of a bipartite graph $G$ of order $N$ and ${\bf v}_m$ the normalized eigenvector corresponding to $\lambda_m$. I am looking for possible ways to evaluate or bound $\sum_{m=1}^N {\bf v}_m(i)/\lambda_m$ for a node $i$ without a priori knowledge of the spectrum and eigenvectors. One may ignore $m$ for which $\lambda_m=0$ in the sum. I am hoping that I can exploit the symmetry of the eigenvalue spectrum and the properties of eigenvectors of bipartite graphs (like the ones mentioned in the answer to this question) to achieve this. Any ideas on how to proceed?

thecorresponding eigenvector? $\endgroup$thein my question. First, you have a whole eigenspace. Second, even if each eigenvalue is simple (which is often not the case), how do you normalize $v$? The bound that you are seeking obviously depends on this normalization. (And, if $\lambda$ is multiple, on the choice of a basis.) $\endgroup$1more comment