  A version of ``preferential attachment'' random graphs, corresponding to
linear ``weights'' with random ``edge additions,'' which generalizes some
previously considered models, is studied. This graph model is embedded in a
continuous-time branching scheme and, using the branching process apparatus,
several results on the graph model asymptotics are obtained, some extending
previous results, such as growth rates for a typical degree and the maximal
degree, behavior of the vertex where the maximal degree is attained, and a law
of large numbers for the empirical distribution of degrees which shows certain
  ``scale-free'' or ``power-law'' behaviors.
