 There are at least two primary ways of thinking in mathematics. The algebraic focuses on numbers, variables, equations, and expressions, while the geometric focuses on drawings and pictures. A good mathematician understands both. A great mathematician blends them. Remember the geometric picture of a graph is actually irrelevant. All that matters are the connections. We might actually refer to this as the topology. This means that we can move the points of a graph and have the same graph as long as the same points are still connected. So we might have two very different looking graphs, but it turns out that they're actually the same. So notice that in both graphs, the vertex labeled 1 is connected to the vertices labeled 2, 3, and 5. The vertex labeled 2 is connected to the vertices labeled 4 and 1. The vertex labeled 3 is connected to the vertices labeled 1 and 5. Vertex 4 is connected to 2 and 5. And vertex 5 is connected to 1, 3, and 4. And since the connections are the same, the graphs are the same. Now it was easy to decide in the example because the vertex labels didn't change. But what if they did? For example, suppose our vertices were labeled this way or this way. Informally we can say that two graphs are isomorphic if we can relabel the vertices of one graph so it has the same edges as the other. If we want a more formal definition, two graphs are isomorphic if there's a bi-checkion between the vertices where uv is an edge in g if and only if the image is an edge in g prime. As with the formal definition of graphs, we won't get back to this for quite some time. The problem is that given two graphs, it may be difficult to find an isomorphism if one even exists. However, if two graphs are isomorphic, they must have the same number of vertices, the same number of edges, and the same degrees. So if we consider the contrapositive, if any of these are different, the two graphs aren't isomorphic. So let's take a look at these two graphs and let's show that they're not isomorphic. So notice that both graphs have five vertices and five edges, but one graph has a vertex of degree 4 while the other does not. And so the two graphs can't be isomorphic. What about these two graphs? So both graphs have the same number of vertices and the same number of edges, and both have vertices of degree 4, 3, and 1, and three vertices of degree 2. So they have the same number of vertices, edges, and their degree sequences are the same so the two graphs must be isomorphic, or are they? Remember if you don't find the flaws in your work, someone else will. Adding two graphs or isomorphic is a tremendous commitment because it's saying two things that look very different are really the same thing. So let's try as hard as possible to find differences between the graphs. Again, the graph is all about the connections. So if we look closely, in one graph the degree 4 vertex is linked to a degree 1 vertex, but in the other graph the degree 4 vertex is not linked to any vertex of degree 1. So the two graphs are not isomorphic. And so we come to the following situation. If two graphs have different numbers of vertices, or different numbers of edges, or different degree sequences, then the two graphs are not isomorphic. On the other hand, if the two graphs have the same number of vertices, the same number of edges, and the same degree sequence, then the two graphs are not necessarily isomorphic. So how can we tell? As usual, the answer is more math.