 We might be able to separate a graph by removing one edge, and any graph can be separated by removing all its edges. A key concept in mathematics and in life, if you can do it at all, there's a first time. And so we define the edge connectivity of a connected graph G, denoted lambda G, is the minimum number of edges whose removal produces a disconnected graph. More generally, if we're not starting with a connected graph, it's the number of edges whose removal breaks off one more component. Note that we may have to remove lambda G-specific edges. For example, suppose our graph is a tree, let's find lambda G. Since every edge of a tree is a bridge, removing any single edge disconnects the graph. So the number of edges we need to disconnect the graph is 1. What if we have a cycle? If we remove a single edge from a cycle, the vertices are still connected. But if we remove a second edge, they are not. So lambda Cn is 2. How about a complete graph? So a useful idea in math and in life, set your boundaries first. Let's find some bounds on our edge connectivity. Now remember, definitions are the whole of mathematics. All else is commentary. The graph we're talking about, Kn, is the graph with n vertices, each of which is connected to every other vertex. Since each vertex in Kn is connected to the other vertices, it must have n-1 edges instant on it. So we can definitely detach the vertex and disconnect the graph by removing these n-1 edges. So lambda Kn is less than or equal to n-1. Could it be less? Well, suppose we want to detach a pair of vertices. We need to have at least three vertices. And we need to remove n-2 edges from each vertex. So we could disconnect the graph by removing 2 times n-2 edges. Now, in order for 2n-4 to be less than n-1, we'd need, but we assume that we had at least three vertices, so in these cases n-1 is always smaller. And so it appears that lambda Kn is n-1. But it's conceivable we might be able to remove fewer edges, so let's prove it. Now, proof is not such a terrible thing. Remember, the purpose of proof is to review concepts, reveal consequences, and raise new questions. And if that's not enough, remember, if you don't find the flaws in your work, someone else will. So let's think about this. Suppose we separate Kn into two parts, g-1 and g-2, by removing our minimum number of edges. Since this is the fewest number of edges we can remove, then our subgraphs should have the maximum number of edges. This suggests they're complete graphs, but are they? Suppose at least one is not a complete graph. It doesn't matter which, so we'll say it's g-1. Since g-1 is not a complete graph, then at some point we've removed an edge in g-1. But since there's something joining vertices in g-1, it can't connect g-1 and g-2. Restoring the edge is the same as removing one viewer edge. So we have a disconnected graph that resulted from removing lambda Kn minus one edges from Kn, but this is impossible. So if we remove the minimum number of edges from Kn, we'll end with two complete graphs, kp and kq. Since we've started with Kn, every point in kp would have originally been connected to every point in kq. So to separate the two graphs, we need to remove q edges from each of p points. Alternatively, we could remove p edges from each of q points, but in either case we need to remove pq edges where p plus q is equal to n. So what's the least value of pq where p plus q equals n? We find the least value occurs when p is n minus one, which separates Kn into Kn minus one at an isolated point. So lambda Kn is n minus one. And again it's worth pointing out proof has value beyond verification. In fact verification should be considered the least important part of proof. In this case, it reveals new consequences. So if we wanted to separate Kn into kp and kq, we need to remove pq edges. And if we didn't separate it this way, we could have separated it more efficiently by removing fewer edges. Perhaps the most important aspect of proof is that proof also raises new questions. And in mathematics and in life, asking questions is important. In this case, suppose g isn't a complete graph but could be separated into kp, kq as disjoint subgraphs. Do we know anything about our original graph? And can we say anything about lambda g? We'll take a look at that later.