The author is using the usual definition where vertices are an arbitrary (finite) set and edges are pairs of vertices. From the book:
Definition 2.1.1. A simple graph is a pair (V, E), where V is a finite set, and
where E is a subset of P_2(V).
Sure in some examples V happens to be a set of natural numbers and then edges must be pairs if natural numbers, but there's nothing weird with that as any set would do. (Incidentally in model theory countably infinite graphs are often assumed to have as underlying set the naturals because it is convenient notationwise)
Definition 2.1.1. A simple graph is a pair (V, E), where V is a finite set, and where E is a subset of P_2(V).
Sure in some examples V happens to be a set of natural numbers and then edges must be pairs if natural numbers, but there's nothing weird with that as any set would do. (Incidentally in model theory countably infinite graphs are often assumed to have as underlying set the naturals because it is convenient notationwise)