An infinite family of cubic nonnormal Cayley graphs on nonabelian simple   groups
  We construct a connected cubic nonnormal Cayley graph on $\mathrm{A}_{2^m-1}$ for each integer $m\geqslant4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on nonabelian simple groups. 
Jiyong Chen, Binzhou Xia and Jin-Xin Zhou
math.CO math.GR