Let $G$ be any graph with at least one edge and let $e$ be any edge of $G$. Let $G-e$ denote the subgraph of $G$ obtained by deletion of the edge $e$. Assume that $G$ has $n$ vertices. Suppose that $\lambda_1(G)\geq\cdots\geq \lambda_n(G)$ and $\lambda_1(G-e)\geq\cdots\geq \lambda_n(G-e)$ are eigenvalues of $G$ and $G-e$, respectively. It is true that $\lambda_i(G)\cdot\lambda_i(G-e)\geq 0$ for all $i=1,\dots,n$?
I think this is false and there are counterexamples.
Here is a sage session.