  We study the relation between the growth rate of a graph property and the
entropy of the graph limits that arise from graphs with that property. In
particular, for hereditary classes we obtain a new description of the colouring
number, which by well-known results describes the rate of growth.
  We study also random graphs and their entropies. We show, for example, that
if a hereditary property has a unique limiting graphon with maximal entropy,
then a random graph with this property, selected uniformly at random from all
such graphs with a given order, converges to this maximizing graphon as the
order tends to infinity.
