Consider the class of graphs on n vertices which have maximum
degree at most n/2-1+τ, where τ≥ -n^^{1/2+ε} for
sufficiently small ε>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.