Decision 1 Maths

Can anybody please explain why Odd Nodes on a Network, come in Pairs please?

Thanks
Vandomo

It's not too complicated, fortunately. Every single arc has two ends so if we add up the valencies of all the nodes in the graph, the total is going to be 2 x number of arcs. Since the sum of valencies is in the form 2n, it must be even. Therefore, if nodes with an odd valency exist in the network, they have to come in pairs in order to make the total valency even. For instance, in the (impossible) situation that we only had one node of odd valence, the total valence would also have to be odd, which have shown to be impossible.

Does that make sense?