Uniform asymptotic expansion for regular graph counts

Let $n$ and $d$ be integers with $1\leq d\leq n-2$ and $dn$ even, and set $\lambda=d/(n-1)$ and $\varLambda=\lambda(1-\lambda)$. Let $\operatorname{RG}(n,d)$ denote the number of $d$-regular graphs on $n$ vertices, and let $p_1,\ldots,p_7$ be the polynomials specified above. Uniform asymptotic expansion for regular graph counts. Uniformly for all such $d$ and $n$,

$$\operatorname{RG}(n,d) = \Big(\lambda^\lambda (1-\lambda)^{1-\lambda}\Big)^{\binom n2} \binom{n-1}{d}^{n} \exp\biggl(\,\sum_{j=1}^7 \frac{p_j(\varLambda)}{\varLambda^j n^{j-1}} + O\bigg(\frac{1}{d^3(n-d)^3n}\bigg)\biggr).$$

This conjecture proposes a uniform approximation for the number of regular graphs across the full permitted degree range, improving earlier finite-term calculations and including an explicit error term.