[Assume all groups are finite]
One knows the general statement that the sum of the values of the character function on the generating set is an eigenvalue of a Cayley graph.
But the above doesn't seem to automatically tell me anything about the ordering among the eigenvalues. Is there an expression or a method of calculating which will directly tell me something about the second largest adjacency eigenvalue of a Cayley graph? (At least when its Abelian?)
Is there way to relate the second largest eigenvalue of a Cayley graph to the length of closed paths and/or the diameter of the Cayley graph or of the volume of of $n-$balls around the vertices? (Again any known simplification for the Abelian case?)