All graphs in this problem are finite and simple.
An embedding of a graph F into a graph G is an injective mapping $\sigma$: V(F) $\rightarrow$ V(G) such that
$\sigma(x)\sigma(y) \in$ E(G) for every edge xy $\in$ E(F). Let emb(F, G) denote the number of embeddings of F
into G. An induced copy of a graph F in a graph G is an induced subgraph of G isomorphic to F.
Let $[G_F]$ denote the number of induced copies of F in G.
Fix k $\ge$ 1 and let G and H be two graphs. Show that the following statements are equivalent:
(1) For every graph F on k vertices, $(\frac{G}{F}) = (\frac{H}{F})$.
(2) For every graph F on k vertices, emb(F, G) = emb(F, H).
(3) For every graph F on k vertices, $[G_F] = [H_F]$.
