I wonder if the following graph problems have been studied and have names.
Problem(s). Given two $n$-vertex unlabeled graphs $G_1$ and $G_2$, find their maximum/minimum edge intersection. That is find two labeled graphs $H_1 = ([n], E_1)$ and $H_2 = ([n],E_2)$ such that $H_1 \simeq G_1$, $H_2 \simeq G_2$, and $|E_1 \cap E_2|$ is maximized/minimized.
It would also be helpful to know if these problems have been studied for special classes of graphs, e.g. for trees.