With my student Marsha Minchenko, we looked for quartic (4-regular) integral (all eigenvalues integers) graphs. We started by looking among Cayley graphs and arc transitive graphs, but then we found some others by taking homomorphic images. Details in a forthcoming paper, but in the meantime, here are the graphs we found:
All graphs are presented in magma format. Arc transitive graphs were originally obtained from the census by Potocnik, Spiga, and Verret.