  There has been much recent interest in random graphs sampled uniformly from
the n-vertex graphs in a suitable structured class, such as the class of all
planar graphs. Here we consider a general 'bridge-addable' class of graphs - if
a graph is in the class and u and v are vertices in different components then
the graph obtained by adding an edge (bridge) between u and v must also be in
the class.
  Various bounds are known concerning the probability of a random graph from
such a class being connected or having many components, sometimes under the
additional assumption that bridges can be deleted as well as added. Here we
improve or amplify or generalise these bounds. For example, we see that the
expected number of vertices left when we remove a largest component is less
than 2. The generalisation is to consider 'weighted' random graphs, sampled
from a suitable more general distribution, where the focus is on the bridges.
