Figure 6

Two non-isomorphic graphs each with five vertices. Note that both graphs have two vertices of degree 3 and three vertices of degree 2. To see that the two graphs are not isomorphic, consider the following: we say a graph is bipartite if the vertices of the graph can be partitioned into two sets such that no edges of the graph have end points within the same partition class. For instance, G₁ is bipartite; consider the partition of its vertices into the sets {v₁, v₃, v₅} and {v₂, v₄}. On the other hand, it is impossible to partition the vertices of G₂ into two such sets. As the property of being bipartite is invariant under permutations of the vertices of a graph, it follows that G₁ and G₂ are not isomorphic.