how to determine if two graphs are not isomorphic What are some good ways of determining if two reasonably simple looking graphs are not isomorphic? I know that you can check their cycle or some weird property (for certain graphs), but are there some other tricks to do this?

With practice often one can quickly tell that graphs are not isomorphic. When graphs G and H are isomorphic they have the same chromatic number, if one has an Eulerian or Hamiltonian circuit so does the other, if G is planar so is H, if one is connected so is the other. If one has drawings of the two graphs, our visual systems are so attuned to finding patterns that seeing that the two graphs have some property the don't share often makes it easy to show graphs are not isomorphic. 
