David Lambert wrote:

> Does the generating function associated with a graph tell us anything interesting about the graph?

Yes! I may be confused, but I think generating function you're talking about is associated with a _pointed_ graph, as explained in my comment above. Anyway, there's a field called [algebraic graph theory](https://en.wikipedia.org/wiki/Algebraic_graph_theory) that studies things like this.

> Does the generating function associated with a graph tell us anything interesting about the graph?

Yes! I may be confused, but I think generating function you're talking about is associated with a _pointed_ graph, as explained in my comment above. Anyway, there's a field called [algebraic graph theory](https://en.wikipedia.org/wiki/Algebraic_graph_theory) that studies things like this.