Graph Theory Problem: why no vertices of order 1?

[Furai]

Hey am not sure if this is the right area to post this topic but i think it is close enough.

Basically i am stuck trying to figure out why if this particular graph has two vertices of degree 5 it cannot have any vertices of order 1.

The graph in question is a simple connected graph with 6 vertices and 9 arcs.

So what i want to know is why this simply connected graph with 6 vertices and 9 arcs (two of the vertices with order 5) cannot have any vertices of order 1

I know that the sum of the order for the graph is 18 but that is it, am not that experienced in this branch of maths

Thanks for your time

 

[ksdhart2]

When I was mulling over this problem, I found it helpful to draw a picture of one possible graph. You are told there are six vertices (or "nodes" as per the terminology I'm familiar with) and 9 arcs ("edges"). Additionally, you're told that two of these nodes have order 5, meaning that 5 edges meet at that node. So try drawing a few graphs with two nodes of order 5. What do you notice about these two nodes? How many other nodes are they connected to? What does that imply about whether there can be a node of order 1 in this graph? Are these characteristics specific to the one graph you drew? Maybe try drawing a different graph and see what you find.