[Math] Does there exist a simple graph that has five vertices each of degree three

Question

Does there exist a simple graph that has five vertices each of degree three?

I say it does not because according to the rule $2m =$ sum of degrees, where $m$ are the edges. The sum of the degrees is $15$, then $2m = 15$. Then, the number of edges $= \frac{15}{2}$, which is not an integer.

Answer 1

As David mentions in the comments, you are correct. The rule you refer to is often called the degree sum formula or the handshaking lemma. The same rule can be used to show the following more general statement:

Any graph must have an even number of odd vertices.

Here a vertex is called odd if it has odd degree.