TITLE: How to show these three-regular graphs on 10 vertices are non isomorphic?
QUESTION [6 upvotes]: The number of vertices and edges are same, with each vertex having the same degree and the degree sequence of the graph is also the same. I have even tried finding a bipartite graph in any one of them even that seems to fail.
Question- How to show the three graphs with degree sequence [3,3,3,3,3,3,3,3,3,3] are non isomorphic (see figure)?

REPLY [2 votes]: The first graph is the only one not containing $4$-cycles, so it's not isomorphic to the other two. The second and third graphs are not isomorphic because the third is planar and the second contains a subdivision of $K_{3,3}$.