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.