f_G of disjoint union

Statement

Let $G, H$ be two graphs on disjoint vertex sets. Then,

f_{G ⊔ H} = x f_{G} (x) f_{H} (x)

where $f_{G}$ is defined here

Proof

Note that a spanning tree of $(G ⊔ H)^{+}$ consists of pairs of spanning trees of $G$ and $H$ . We multiply by $x$ as for each graph, the degree of 0 will be

\deg_{T_{G}} (0) + \deg_{T_{H}} (0) - 1 = 1 + \deg_{T_{G}} (0) - 1 + \deg_{T_{H}} (0) - 1