A Cayley graph can be consider as an interpretation of group representation. This means that one can construct the Cayley graph Г = Cay(G, S) by having the vertex set to be the elements of G with g~h if and only if hg-1 ε S. In this paper, we focus our attention on Cayley graphs with a normal symmetric subset. We also determine all eigenvalues of Buckminster fullerene by constructing Cay(G,S) where G is the alternating group A5.