If a graph has

8 verticies:

2 with 1 degree

1 with 2 degrees

5 with 3 degrees.

If it is not a tree I must prove why.

The only way I can think of to do this is to draw the graph (which I can't replicate here).

When I draw the graph I find I have 2 non-trivial circuits, therefore the graph is not a tree.

Is this the only way to prove the graph is not a tree?