What does small eigenvalue gap imply for a graph?

Question

Knowing a graph has good expansion has well-known implications. What can we say about graphs with $1-\lambda = O(\log n/n)$, where $1-\lambda$ is the difference between the two largest eigenvalues of the graph? Can such an equation hold for a graph that locally expands very well but has some small badness?

Answer 1

One result related to this is Cheeger's Inequality, which gives a partial converse to the statement "large eigenvalue gap implies good expansion". Roughly speaking, it says that if the spectral gap is small, there is at least one cut with poor conductance (few edges crossing the cut relative to the volume of the smallest part).

I don't believe you can say much about local expansion just from knowing the eigenvalue gap -- for example, a graph consisting of two disjoint complete graphs on $n/2$ vertices has $1-\lambda=0$, but all small sets expand well.