The generating graph of a profinite group

Let G be a 2-generated group. The generating graph Γ (G) of G is the graph whose vertices are the elements of G and where two vertices g and h are adjacent if G= ⟨ g, h⟩. This definition can be extended to a 2-generated profinite group G, considering in this case topological generation. We prove that the set V(G) of non-isolated vertices of Γ (G) is closed in G and that, if G is prosoluble, the graph Δ (G) obtained from Γ (G) by removing its isolated vertices is connected with diameter at most 3. However we construct an example of a 2-generated profinite group G with the property that Δ (G) has 2ℵ0 connected components. This implies that the so called “swap conjecture” does not hold for finitely generated profinite groups. We also prove that if an element of V(G) has finite degree in the graph Γ (G) , then G is finite.