Suppose that $G = ([2n], E)$ is a $d$-regular, undirected graph. If every subset of exactly $n$ nodes has at least $d^2$ edges emanating away from it (so $E(S, \overline{S}) \geq d^2$ for every $S \subset [2n]$, $|S| = n$), then what can you say about the second largest (in absolute value) eigenvalue of the adjacency matrix?
Another way to ask this question is if I know what the min bisection size is of my $d$-regular graph above, what can I conclude about the second largest (absolute) eigenvalue?
(For example, if you could say that the eigengap is large, it would imply that if the eigengap is small, there'd be a cut in the graph of size at most $d^2 - 1$.)