The matrices representing graph structure are fundamental to graph algorithms. Many such algorithms rely on low-rank approximation and thus focus on eigenvalues and eigenvectors with large absolute values. Low-rank approximation seeks the closest low-rank matrix to a given matrix; its optimal solution is obtained by sequentially selecting the eigenvalues and eigenvectors with the largest absolute values. This suggests that eigenvalues and eigenvectors with large absolute values capture most of the original matrix’s information and are therefore deemed important. However, adding the same value to all diagonal entries uniformly shifts every eigenvalue, so assessing importance based solely on absolute value can be misleading. In this paper, we incorporate Graph Spectrum Shift (GSS) into low-rank approximation to identify eigenvalues and eigenvectors whose importance does not depend on their original absolute values. Numerical experiments on synthetic graphs and realistic graphs using the adjacency matrix, the Laplacian matrix, and the normalized Laplacian matrix demonstrate that low-rank approximations with GSS often select eigenvalues with smaller absolute values when constructing the optimal low-rank matrix.