 After mathematicians define an object, they usually introduce some operations on that object. So what can you do with the graph G? Well, start with unary operations, things you can do with just a single graph G. Given the graph G, the complement of G, indicated this way, is the graph with vertex set and edge set, where the vertex set is the same as the vertices of the original graph, and E is in the edge set of the complement, if and only if E is not in the edge set of the original graph. So the complement has the same vertices and edges between the non-adjacent vertices of our graph. So if we want to find the graph complement, since the vertices are the same, we only need to join the non-adjacent vertices in our graph to produce the edges of our complement. So let's join vertices that don't have an edge between them, then we'll remove all the edges that were in the original graph. And remember, our vertices stay the same, so even though we get an isolated vertex, it is in the complement. Another thing we could do with a single graph is to find the power of the graph. The square of G, designated G squared, consists of the same vertices V, and an edge set E, where UV is an edge in G squared, if the distance between U and V is less than or equal to 2. G cubed, G to the fourth, and so on are defined similarly. So for the graph shown, find G squared and G cubed. So G squared includes all edges of G, to which we'll add edges between two vertices in G, separated by two edges. So we'll join this vertex and these vertices, which are two edges away, this vertex to vertices, which are two edges away, and so on. And here these are added edges. We still have all the original edges. And so our graph looks like for G cubed, we'll include edges between vertices separated by three edges in G. So from here to the third vertex out, and from this vertex to the third vertex out, and so on, and so on. Now, remember that given a graph G, we can form a subgraph by selecting some edges and their incident vertices, and possibly some extra vertices. An edge-induced subgraph is formed by selecting some edges and their incident vertices only, and an induced subgraph is formed by selecting some vertices and all their incident edges. We can also view the process of forming a subgraph as the removal of some vertices and edges. If we remove a single vertex V and all its incident edges from G, we indicate this as G minus V, and if we remove a single edge from G, we indicate this as G minus E. We could also remove a set of vertices or a set of edges, but we usually focus on removing just one edge or just one vertex. For example, from the graph shown, let's find G remove V2 and G remove E3. So note that the vertex V2 is incident on the edges E1, E7, E5, and E6. So if we remove V2, we must also remove those edges. And that gives us the graph G minus V2. Restoring the original graph, if we remove edge E3, we only remove the edge. Well, now, what if you have two graphs, G1 and G2? There are several binary operations on the graphs. We'll assume that the vertex and edge sets are disjoint. The union of two graphs, G1, G2, is defined in the obvious manner. As it's a set operation, there isn't really much we can do with it. So for the given graphs, find the union. The vertex set for the union is the set of all vertices in both graphs. The edge set for the union is the set of all edges in both graphs. So the union is, wait for it, both graphs. Again, set operations tend not to be that exciting. On the other hand, the join of two graphs is the set of all edges and vertices in the two graphs. And in addition, we'll introduce a new set of edges, those joining a vertex in G1 with a vertex in G2. We designate the join G1 plus G2, except when we don't. We'll see why I introduced that qualifier in a moment. So let's find G1 plus G2. We'll start with the union. We'll do this quickly, so watch carefully. The two graphs, our union, will be the same two graphs. But since this is a join, we'll also add edges between every vertex in G1 and every vertex in G2. So from this vertex to the vertices of G2, and from the others, and there's our join. Now, if G1 consists of N isolated points and G2 consists of M isolated points, we use the notation KNM for the join of these two sets of isolated points. For example, if we want to draw K34, K34 consists of a set of three points and a set of four points, where every point in the first set is joined to every point in the second set. And so that looks like...