a quick minute to talk about the relationship between the degrees of vertices and edges in a graph so here I have a very very simple graph two towns and a road in between them now this has a degree of 1 it has one edge going into that vertex I'm just going to write degree one there this one has degree 1 so all together in this graph we have degree 1 plus degree 1 we have two total degrees we add the vertices together how many edges do we have we have one road ok let's just add a little bit of complexity to this thing we'll add in another vertex we'll add in a road over here and we'll add in a loop there ok now let's consider the degrees this one has degree 2 now this one has degree 1 still I should have rub that out and this one has degree 1 2 3 all together I have 2 + 3 + 1 is 6 6 total and edges 1 2 3 all right let's add some complexity to this and a line and the line another line there so I've got a loop I've got multiple edges let's figure out the degree of all of our so this one has degree 1 2 3 4 this one has degree 1 2 3 this one has degree 1 2 and this one has degree 1 2 3 and altogether that's 4 plus 3 plus 3 plus 4 plus 3 plus 2 is 12 ok and how many edges do I have well 1 2 3 4 6 okay hopefully you can see a pattern starting to emerge and this pattern is always going to work because an edge moves from vertices vertice so every edge always has two vertices attached to it and we get this so the relationship I hope you're seeing is 1 2 3 6 6 12 the total degrees in a graph is always double the total number of in a graph the sum of the degrees of the vertices in a graph so the sum add up the degrees of the vertices in graph is double the number of edges