I want to ask the question in two parts,
(1)
Is there some fundamental distinguishing property between Abelian and non-Abelian Cayley graphs? (say some specific proof technique which distinguishes them?)
(2)
Are there any set of (constant degree) (Abelian) Cayley graphs which are expanders? Do they have any distinguishing property?
For reference one can see chapter 5, starting on page 30 of these notes, http://www.eecs.berkeley.edu/~luca/books/expanders.pdf
Best Answer
To add to Anthony's comment, one can make an explicit connection between the large number of walks between vertices and the spectra of Abelian Cayley graphs. It turns out that constant-degree Abelian Cayley graph are not only bad expanders,but they tend to be disconnected (they have a positive proportion of their eigenvalues close to their valency). See: http://www.math.udel.edu/%7Ecioaba/inpress_version.pdf for a short proof. Alon and Roichman showed earlier that Abelian Cayley graphs have large diameter (power of the order of the graph); the diameter of an expander should be logarithmic in the order of the graph. A proof (using character sums) that the Abelian Cayley graphs have large 2nd eigenvalue is given by Friedman, Murty and Tillich: http://www.mast.queensu.ca/~murty/f-m-t.pdf