From Paulus Graphs.
“The (25,2)-, (25,4)-, and (26,10)-Paulus graphs have the apparently rather unusual property of being both integral graphs (or asymmetric) and identity graphs (a graph spectrum consisting entirely of integers).”
I once quietly conjectured this was impossible, but I was proven wrong. Very wrong. Eric Weisstein was amused enough by my reaction that he added the above quote.
Is there a smaller counterexample?