You may want to look at the work of Arora, Barak, Ragavendra, Steurer etc relating details of graph spectra to expansion of small sets and to the Unique Games Conjecture.

You may want to look at the work of Arora, Barak, Ragavendra, Steurer etc relating details of graph spectra to expansion of small sets and to the Unique Games Conjecture. Sample references are:

Ragavendra–Steurer, Ragavendra–Steurer–Tetali,Arora–Barak–Steurer.