Lets say I have two graphs $G_1$ and $G_2$ which are defined on a disjoint set of vertices, $V(G_1)$ and $V(G_2)$, such that there is no edge connecting any $V(G_1)$ and $V(G_2)$. So basically I have two disconnected graphs.
Is there any way of relating the eigenvectors &/or eigenvalues of $G_1$ and $G_2$ to the graph obtained by adding one edge that connects $V(G_1)$ and $V(G_2)$ or in the converse direction?
Are there any general or special case (e.g. for regular graphs) results on this?