A Markov chain approach to the study of randomly grown graphs is proposed and applied to some popular models that have found use in biology and else where. For most randomly grown graphs used in biology, it is not known whether the graph or properties of the graph converge (in some sense) as the number of vertices becomes large. Particularly, we study the behaviour of the degree sequence, that is, the number of vertices with degree 0, 1,..., in large graphs, and apply our results to the partial duplication model. We further illustrate the results by application to real data.