Universal graphs at the successor of a singular cardinal

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of colinality N-0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such mu there are mu(++) graphs on mu(+) that taken jointly are universal for the graphs on mu(+), while 2mu(+) much greater than u(++). The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal kappa exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.