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 is supercompact and the cardinals θ > 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 () =, the Singular Cardinal Hypothesis fails at κ and there exists a universal graph at cardinality λ ∈ (κ, 2 κ) (, 2^).