Checking for Graph Isomorphism

1. Different Number of Vertices. If two graphs have different numbers of vertices, they cannot be isomorphic.
   Example: One graph has 4 vertices and the other has 5 vertices.
2. Different Number of Edges.
   Example: Graph \(G_1\) has 6 directed edges, while \(G_2\) has 7 directed edges.
3. Different Degree Sequences. [Recall that the degree sequence of a graph is the list of vertex degrees arranged in sorted, non-increasing order.]
   Example: The following graphs feature different degree sequences; one has $(3, 1, 1, 1)$ and the other has $(2, 2, 1, 1)$.
   

   The graph $G_1$ with the degree sequence $(3, 1, 1, 1)$. Miriam Briskman, CC BY-NC 4.0.

   

   The graph $G_2$ with the degree sequence $(2, 2, 1, 1)$. Miriam Briskman, CC BY-NC 4.0.