McKay–Wormald's asymptotic enumeration conjecture for graphs with given degree sequence

Let ${\bf d}= (d_1,\ldots,d_n)$ be a degree sequence, let $g({\bf d})$ denote the number of graphs with degree sequence ${\bf d}$, let $d=\frac{1}{n}\sum_{i=1}^n d_i$, let $\mu=d/(n-1)$, and let $\gamma_2=(n-1)^{-2}\sum_{i=1}^n(d_i-d)^2$. McKay–Wormald's conjecture. For some absolute constant $\varepsilon>0$, if ${\bf d}={\bf d}(n)$ satisfies

$$\max_j|d_j-d|=o\left(n^\varepsilon\min\{d,n-d-1\}^{1/2}\right),\qquad n\min\{d,n-d-1\}\to\infty,$$

and $\sum_i d_i$ is even, then

$$g({\bf d})\sim \sqrt{2}\exp\left(\frac14-\frac{\gamma_2^2}{4\mu^2(1-\mu)^2}\right)\left(\mu^\mu(1-\mu)^{1-\mu}\right)^{n(n-1)/2}\prod_i\binom{n-1}{d_i}.$$

This conjecture unifies asymptotic formulae known in sparse and dense ranges and extends them to the intermediate range for degree sequences sufficiently close to regular; the paper proves the conjecture, so the enumeration formula is now established under these hypotheses.