 Okay so in graph theory today I'm going to just talk about two types of graphs I'm actually going to just be about Mathematica today and I'm going to show you some Platonic graphs. Remember those are the tetrahedral, octahedral, cubical, icosahedral, dodecahedral graphs. I'm going to show you what they look like and then I also want to show you the Peterson graph. I just want to take the Peterson graph. Now you know what's happening in graph theory. You've seen a couple of lectures now and it's just graph after graph after graph. Well I like to introduce graph theory in this way because you've just got to get familiar with all of these things and what we're doing is we just have this box and we're just piling stuff in this box and later on when we're so familiar with them it becomes easy just to drag them out of that box. They're just part of your mind and you just boom boom they come out and you can just use these in graph theory. So very interesting to look at these start developing the concepts, start learning the language. It's like a child learning a brand new language or an adult learning a brand new language. It's become familiar. Just enjoy these graphs at the moment and just get them in your head and later on they'll just pop out and you'll just be able to talk this language and use these things when we do some calculations or do some something more interesting with these graphs. So for now just absorb all of these interesting graphs and learn Mathematica, learn how you can use Mathematica, the power of Mathematica and play around with these and develop your own graphs or look up to various ones. So this one let me show you some great graph data inside of Mathematica and that's going to be very interesting for you to come up with and explore new graphs, explore this wonderful world of graphs. For now just pump your head full of this interesting stuff. So here we are in Mathematica. Let me show you these platonic graphs first of all tetrahedral, octahedral, cubical, icosahedral and dodecahedral graphs. Very beautiful. Let's look let's see what they look like and then just explore one or two things about them. So let's do all of them in one go. So for that I'm going to do the table, use the table function and I'm going to use the keyword graph data. Graph data and I'm going to use a placeholder called n comma and now let's specify what n is going to be. Now I'm not going to let n go from one to four to five or whatever. I'm going to let it cycle through some nominal data, some strings. So I'm going to put a comma and put all of those inside of square brackets. So the first one I want is tetrahedral graph. Tetra, now I must not make any spelling mistakes here because there's no autocompletion. Tetrahedral graph. Now I am not the world's best typist so if there's a mistake in here I'll correct it in post-production but let's go let's go with it. Octahedral, octahedral is, wait octahedral, there we go. Octahedral graph, remember to put the graph in the uppercase G there and then we had the cubical graph. There we go, cubical graph. The next one was the icosahedral graph, icosahedral graph and then the most difficult one of all the dodecahedral graph. Let's go for that one, dodecahedral, dodecahedral. This is very funny. Anyway, graph, there we go and I've got to close my curly braces so you see these are the strings and strings go inside of quotation marks. These are well these are all just nominal data types so it's this one, two, three, four, five of them, they're in a list so they go inside of curly braces but it's what N is cycling through. So N is going to cycle through first tetrahedral group then octahedral group, then cubical graph, then icosahedral graph, then dodecahedral graph and they are all going to be printed as graph data and this is all part of a table function so let me close that. We see that the Wolfram Research Data Server has been queried here in the desktop version and it's going to download this graph data from the Wolfram Research Data Server which might take a while depending on your internet access and how busy their servers are etc. And there we are, lo and behold, we have a tetrahedral graph, octahedral graph, a cubical graph, icosahedral graph and a dodecahedral graph. Okay let's just explore some of these. Let's just have a look here at this first one. Let's just do the tetrahedral graph. We see first of all that it's a planar graph and remember we did planar graphs before, we're going to get back to planar graphs, they're very important and look at this. If I look at the edge count of this top little vertex or node here, I note that it's 1, 2, 3. Let's look at this one down here, it's 1, 2, 3. Let's look at this one here, it's 1, 2, 3 and let's say the central one, 1, 2, 3. So this is a regular graph. Let's look at the octahedral graph. 1, 2, 3, 4. 1, 2, 3, 4. 1, 2, 3, 4. 1, 2, 3, 4. 1, 2, 3, 4. 1, 2, 3, 4. There we go. Explore these, look at them in a bit more detail. Of course just looking at them as they are drawn here, they are planar graphs. But please explore the number of degrees of each of these edges. Think about the things that we've learned before and the terms that we might have used before as far as these platonic graphs are concerned. Let's just look at the Peterson graph specifically and I'm going to use graph let's do this alt or command option key I should say on a Mac, alt or option and 4 and we're going to say the Peterson graph. There we go. Arrow down and let's just do this. Again we're going to find it using the graph data so that is querying the Wolfram servers just to get this information and you see a few come up there already. Look at all of these and it's not nearly all of them but look at this list of graphs that you can explore. Please explore them. Be adventurous. It's fantastic. Peterson graph is the one that I want. Let's have a look at that. Again it's going to install this from the data servers and it's going to take a couple of seconds and there we go the Peterson graph. As we look at it here is this a planar graph? Is it possible? Let's just look at the edge count. One two three, one two three, one two three, one two three, one two three, one two three, one two three, one two three, one two three. That seems to be all the same. Now have a look at this. Let's do graph data again. I'm going to say graph data. I'm going to say the Peterson graph please. Peterson graph and I'm going to use a second argument which is called alternate names. Alternate names. That's new. Look at that. 10 cubic graph 19. 10 cubic hamiltonian graph 1. 10 edge transitive graph 13. So many interesting tantalizing things to come. Look at all those alternate names for the Peterson graph and it all means something wonderful. Okay play with these. Play with graph data this time around and figure out some things about the platonic graphs with the information that is inside of your head right now. Play around with Mathematica. It's fantastic.