I am looking for examples/families of graphs with the following (maybe vague-sounding at first) property: the graph $G$ has a relatively large subgraph $B$ such that $B$ is bipartite-plus-a-few-edges and the cut between $B$ and $G-B$ is small.

In a paper called A characterization of the smallest eigenvalue of a graph (J. Graph Theory, 1994) Desai and Rao studied this problem and related it to the signless Laplacian spectrum. What I am after here is locating some explicit examples.

EDIT: I can construct lots of examples by just taking a bipartite graph and loosely attaching it to another large dense graph, and then sprinkling some noise. But I'd like to know if there are well-known interesting graphs/families that already display such a behaviour.

EDIT2: Let's take a concrete example. Do the Paley graphs have such a property?

EDIT3: Just a bounce, since the bounty expires tomorrow...