Let $L$ be the Laplacian matrix of a simple, connected graph, and $\mathcal{P}_j$ the projector into the vertex $v_j$, represented by the appropriate canonical basis vector $(0,...,1,...,0)^T$. Given the positive real parameters $t$ and $\lambda$, consider the functions $$P_j(n,t,\lambda)=v_j^T e^{-it(\lambda L-\mathcal{P}_j)}s, $$ where $s=\frac{1}{\sqrt{n}}(1,...,1)^T$ is the normalized uniform vector.

My objective is to show that the condition $P_i=P_j$ implies the degrees of the two vertices $v_i$ and $v_j$ are the same. Does anyone have any tips on how to accomplish this?