 We'll start with graphs for the nodes, points or vertices, and edges, lines or links, are undistinguished. In other words, the only features of a graph are the points and the edges connecting them. Later on we'll allow the nodes and edges to have weights and even directions, but for now we'll keep things simple. We'll also assume no edge connects the vertex to itself, and at most one edge connects two vertices. This gives us a simple, unweighted graph. The trivial graph has one vertex and no edges, and the graph is totally disconnected if it has no edges. These graphs are important as limiting cases, but they're not very interesting or useful. One important idea is that a graph is connected if there is a path between any two vertices. Otherwise, the vertices and edges can be partitioned into connected components. For example, we might try to find how many connected components we have in this graph. And if we look carefully, we see that, well, these four points form a connected component, but you can't navigate from any one of these four points to any one of the three remaining points. Notice that even if we have the drawing of a graph, deciding whether it's connected can be difficult, and if the graph has thousands of links and nodes, the picture will be almost unreadable. An important type of graph is known as a complete graph. A complete graph on N vertices, typically written KN, is a graph where an edge exists between any two vertices. So let's find K4, that's four vertices, and we want to make sure that every vertex is connected to every other vertex. So we'll make a connection between vertex one and vertices two, three, and four, and then from vertex two to the other vertices, and from vertex three to the other vertices. As a combinatorial question, we might ask how many edges does a complete graph have? Since an edge must join any two vertices, then in K5 there are five vertices, pick two of them, and so there will be ten edges in K5. A graph is K regular if every vertex has degree K. We also say that these graphs are regular of degree K, and it should be clear that KN is N-1 regular. Let's see what these graphs look like. Let's try to draw a two regular graph with five vertices. So we know how many vertices we have, so we'll put down five vertices. In order to be two regular, each vertex must have two edges incident on it, and every edge must join two vertices. So if we start here, we can join the two adjacent vertices, and now we have to join this vertex to something else, and so on, and we can form a graph that is regular of degree two by joining every vertex to its, so to speak, adjacent vertex, where we put adjacent in quotes because remember the spatial location of the vertices is actually irrelevant. So these vertices are adjacent only in the non-grafting sense. This does suggest a special type of graph. A cycle, often written as ZN, is a two regular graph with N vertices. Visually the vertices of a cycle form a loop. But remember, the graph only consists of the vertices and the edges. The spatial locations are irrelevant, and so a cycle might not look like a loop. So two regular graphs are cycles. What about three regular graphs? Let's find a three regular graph with six vertices, if it even exists. So we'll start with six vertices, six vertices, and to begin with, we'll begin with the vertex and join it to three others. To proceed, there are two useful ideas in math and in life. It's easiest to work from blank slate, and it's easy to go where you've gone. What this means is that while we could try and work with some of these other vertices and make them of degree three, it's easiest if we start with a blank slate. So let's take one of the unused vertices and join it to the three others we've already connected. So there, we're starting with a blank slate, but we're also going someplace we've already gone. Now, note this increases all their degrees by one, so they all have degree two now. So we join the third unused vertex to the other three. All their degrees will be three, and we'll get our three regular graph with six vertices.