I am reading the following paper:
"Y. Fan, On Spectral Integral Variations of Graphs", in particular, the following part:
I am confused about the following part:
If $\lambda_2(G) = \lambda_2(G+e)$, then $v_2^{(i)}=v_2^{(j)}$, $v_2^TL(G)v_2 = \lambda_2(G)$.
It seems not correct to me since $v_2$ may not be the eigenvector of $L(G)$. For example, the cycle graph with $N=5$. The second eigenvector of $L(G+e)$ is not the same as the second eigenvector of $L(G)$.
Am I correct? or does anyone know what the author really want to say about this?