I have a "Which of the following graphs are isomorphic?" question. The graphs are the same order and regular, which makes it quite difficult for me to prove why they are isomorphic, or find out why they are not. One thing I recognized is that, one of them is 2 colorable whereas the other one is not. Please prove that chromatic number is invariant under isomorphism, so that I can prove these two graphs are not isomorphic.
I am still struggling with the other two graphs, so my other question is: if the order and degree sequence are the same; what could be other invariant that will help me to prove they are not isomorphic (such as chromatic number helping me)? If everything seems to be the same, what is the formal proof? I suppose showing that the two graphs have equivalent adjacency matrices?