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.