 Graph theory is useful for describing relationships. One common type of relationship is a hierarchy. Organization flowcharts, sorting diagrams, succession plans. Consider a hierarchy. One important feature of a hierarchy is that there are no cycles. We define a tree is a connected graph with no cycles. Let's see where this takes us. Suppose G is a tree. Since it's connected, there's a path between any two vertices, U and V. Could there be a second path? If there was, we could form a cycle. But trees don't have cycles. And so suppose G is a tree. Then between any two vertices, there is a unique path. One way to really understand mathematics is to follow every possible consequence of a result. We can do so as follows. Suppose we start with a conditional if A then B. We can link this to a conditional if B then C, which we can then link to a conditional if C then D. How far can we go? So the theorem we just proved concluded that between any two vertices in a graph, there is a unique path. So suppose that in a graph G with N vertices, there is a unique path between any two points. Then we know G has no cycles, because again, if there's a second path, there must be a cycle. And because there is a path, G is connected, and we know that a connected graph with N vertices has to have at least N minus one edges. So we know our graph has N minus one edges. It could have more edges, but we know it has N minus one edges. And consequently, if there is a unique path between any two points, G has N minus one edges and no cycles. Now, it's important to notice that while we proved that G has N minus one edges and no cycles, we did not prove that G is connected. So what if G only had N minus one edges and no cycles, and we didn't know whether or not it was connected? Well, we actually proved something about this. We do know that G is connected. But if G is connected with N minus one edges, we know something about that as well. Adding one edge introduces exactly one cycle. And again, let's start with our consequent. G has a graph with N vertices and N minus one edges, and suppose that adding one edge introduces one cycle. And again, we don't know at this point if G is connected. So let's consider two vertices, U and V. If we add an edge between them, we create a cycle. But this means that without the edge, a path had to exist between U and V. So the original graph had to be connected, and we have... Suppose G has a graph with N vertices and N minus one edges. If adding an edge produces a cycle, then G is a connected graph with N minus one edges. So again, starting with our consequent, G is a connected graph with N minus one edges, then we know that G is connected and every edge is a bridge. And since every edge is a bridge, the graph can have no cycles. So G is a connected graph with no cycles, which means that G is a tree giving us a theorem, which puts us right back at our starting point. Now we've produced a sequence of conditionals if A then B, if B then C, if C then D, and so on where the last conditional takes us back to our starting point if Z then A. And what this means is that we can start with any statement and conclude any other statement. It's almost as if there's a graph here. So for example, suppose that G is a graph with N vertices. Let's prove that if G has N minus one edges and no cycles, then there is a unique path between any two vertices. So remember, we can always assume the antecedent, the if portion. So suppose G has N minus one edges and no cycles, we know G is connected. But since G is a connected graph with N minus one edges, every edge in G is a bridge. And if every edge in a connected graph is a bridge, then G is a tree. But if G is a tree, there is a unique path between any two vertices.