Universal graphs between a strong limit singular and its power

The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of G C H GCH κ κ is supercompact and the cardinals θ > κ θ > κ , λ > κ λ > κ are regular, as an application of a more general method, we obtain a forcing extension in which c f ( κ ) = θ cf(κ ) = θ , the Singular Cardinal Hypothesis fails at κ κ and there exists a universal graph at cardinality λ ∈ ( κ , 2 κ ) λ ∈ (κ ,2^κ ) .