TITLE: Cayley graphs of $A_n.$
QUESTION [17 upvotes]: Consider the Cayley graphs of $A_n,$ with respect to the generating set of all $3$-cycles. Their properties must be quite well-known, but sadly not to me. For example: what is its diameter? Is it an expander family? (I assume not) Is there a nice description of the eigenvalues? The tree number?

REPLY [13 votes]: For $n\geq 5$, all the eigenvalues of the Cayley graph you describe are integers.  The eigenvalues correspond to character values on $3$-cycles multiplied by the number of $3$-cycles (because the generating set of $3$-cycles is a conjugacy class). The multiplicity is the dimension of the irreducible representation.
Since the character values are algebraic integers, it is enough to observe that on a $3$-cycle all character values are rational. This follows from the well-known fact (see Noam Elkies answer to this question) that if $g$ is an element of a finite group $G$, then $\chi(g)$ is rational for all characters $\chi$ of $G$ if and and only if $g^m$ is conjugate to $g$ for all powers $m$ coprime to the order of $g$.  
Since every power of a $3$-cycle that is not the identity is a $3$-cycle and $3$-cycles are all conjugate in $A_n$ for $n\geq 5$, this gives the integrality of character values on $3$-cycles.