Does a graph with $0$ vertices count as simple? Does a graph with $0$ vertices count as a simple graph?
Or does a simple graph need to have a non-empty vertex set?
Thanks!

It is typical to refer to a graph with no vertices as the null graph. Since it has no loops and no parallel edges (indeed, it has no edges at all), it is simple.
That said, if your present work finds you writing "Such and such is true for all simple graphs except the null graph", then it could be a good idea to announce at the beginning of your document that you will not consider the null graph to be simple.
