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:

- A complete catalogue of bipartite quartic Cayley integral graphs.
- A complete catalogue of bipartite quartic arc-transitive integral graphs (that are not Cayley).
- A complete catalogue of quartic Cayley integral graphs (that are not bipartite).
- A complete catalogue of quartic arc-transitive integral graphs (that are neither bipartite nor Cayley).
- A partial catalogue of bipartite quartic integral graphs (that are neither arc-transitive nor Cayley).
- A partial catalogue of quartic integral graphs (that are not bipartite, arc-transitive or Cayley).

All graphs are presented in magma format. Arc transitive graphs were originally obtained from the census by Potocnik, Spiga, and Verret.