Diameter of PA random graphs with edge-step functions

In this work we prove general bounds for the diameter of random graphs generated by a preferential attachment model whose parameter is a function f:N ->[0,1] that drives the asymptotic proportion between the numbers of vertices and edges. These results are sharp when f is a regularly varying function at infinity with strictly negative index of regular variation -gamma. For this particular class, we prove a characterization for the diameter that depends only on -gamma. More specifically, we prove that the diameter of such graphs is of order 1/gamma with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of slowly varying functions are also obtained.