 So let's continue exploring a few more graphs or more concepts. I think when you start off in graph theory, it's better just to have a look at a couple of these and get a couple of concepts. It's just fun to do and then later on we can get to the real mathematics of the power that at least is in graphs. And what the concept I want to talk to you about today is called a subgraph. What is a subgraph? Well I mean the name says it all. Imagine that I have a bunch of vertices and there's an edge there and there's an edge there and there's an edge there. And just imagine that there are more nodes and there's an edge and there's an edge and there's an edge and there's an edge. That's another graph but this original one is part of this bigger one making it a subgraph. So a subgraph of a graph is just a graph which uses some of the edges and some of the nodes in the original graph. Now the whole graph is itself is a subgraph of itself. This is a weird concept but it is. And so a subgraph to see that is we've got a graph and we get to use some of the nodes and some of the edges in that and you can well imagine that there are many subgraphs. You know if you have a large complex graph there are many subgraphs inside of a graph. Let's go to Mathematica and I'll show you a beautiful way to illustrate what a subgraph is. Let's go. There we go. Let's illustrate through the use of the Wolfram language inside of Mathematica here what a subgraph is all about. I'm going to create a graph called G. It's a computer variable name. Remember a computer variable name is just this name that you give to this reserved space in your computer's memory and inside of that you're going to put an object. In this instance I'm going to put a complete graph. That's my object that's going to go inside of this space in memory that's called G. That G is a computer variable and within certain limits you can call it whatever you want. And this for me is going to be a complete graph and I'm going to use 10 nodes or vertices and I want the vertex labels to be visible. So I'm just going to say I want their name please and a beautiful beautiful we see there a very beautiful complete graph. Node a new term that I want you to use to I don't think I've mentioned before is just adjacency. So any node that is connected to another node by a direct edge those two nodes are adjacent. So adjacency exists all along this complete graph because every vertex is connected to every other vertex. I'm going to create a new computer variable called G sub 1 and that G sub 1 I want it to be a sub graph. So I'm going to use the sub graph keyword and the first argument is the graph from which I want it to be taken. It's going to be taken from G and I'm going to say use vertices 1 2 3 and 4 and this is have the vertex labels vertex labels name. There we go. Close our square brackets and there we go. It is a sub graph so it's going to take all of the vertices 1 2 3 4 those 4 that I've specified and all the vertices that are in the edges that are involved with those just those four vertices. And how would this fit in? Let me show you by this wonderful keyword this wonderful function called highlight graph. Highlight graph. There we go. First argument is the graph itself. The second one is the sub graph that I want and then I'm going to do this graphics highlight style or graph highlight style I should say graph highlight style and I don't want one of those I want my own one which is thick is one of the options it won't won't be shown but there's quite a few and I'm going to say vertex labels of course I just want the name just in case you're wondering if if using the online version and you don't get these little arrows let me just show you it's just the minus of course and the greater than sign as I said and then I can type name and did you see how my minus greater than sign turned into a little arrow it's as simple as that and look at the beauty that we have here there's one two three and four and we can clearly see it is a sub graph of our larger graph so play around with these create a few graphs and play around with creating some sub graphs use different graphic highlight styles and see what you can come up with