Consider the class of graphs on $n$ vertices which have maximum
degree at most $\half n-1+\tau$, where $\tau\geq -n^{\half+\epsilon}$ for
sufficiently small $\epsilon>0$.  We find an asymptotic formula for the
number of such graphs and show that their number of edges has a normal
distribution whose parameters we determine.  We also show that
expectations of random variables on the degree sequences of such graphs
can often be estimated using a model based on truncated binomial
distributions.