You disprove isomorphism between two graphs most easily by finding some property that distinguishes them, but an isomorphism would preserve. For instance, the number of nodes of any given degree, whether two nodes of some specified degrees are neighbours, the length of the longest path, and so on.
Once you've used a few of these properties to rule out a lot of potential isomorphisms, it's time to try to prove that the remaining ones are isomorphic. One way to prove that two graphs are isomorphic is to number the nodes of one graph, and down all neighbouring pairs of numbers you get. Then see whether you can in any way number the nodes on the second graph so that you get the exact same pairs of neighbouring numbers.
You disprove isomorphism between two graphs most easily by finding some property that distinguishes them, but an isomorphism would preserve. For instance, the number of nodes of any given degree, whether two nodes of some specified degrees are neighbours, the length of the longest path, and so on.
Once you've used a few of these properties to rule out a lot of potential isomorphisms, it's time to try to prove that the remaining ones are isomorphic. One way to prove that two graphs are isomorphic is to number the nodes of one graph, and down all neighbouring pairs of numbers you get. Then see whether you can in any way number the nodes on the second graph so that you get the exact same pairs of neighbouring numbers.