Verification of the eighth- and ninth-order coefficient polynomials

Prove the conjectured formulas for the coefficient polynomials p_8(x) and p_9(x) in the asymptotic expansion for the number of labelled d-regular graphs.

Background

The authors compute the coefficient polynomials p_j(x) through p_7 using cumulant expansions and compare the resulting asymptotic formula with exact numerical counts. Agreement with an independent expansion enables them to propose explicit formulas for p_8(x) and p_9(x).

These formulas are presented as a conjecture rather than as proved consequences of the paper's expansion. The numerical table later in the section explicitly assumes the conjecture when using the k = 8 and k = 9 approximations.

References

In fact, this enables us to conjecture the next few coefficients in our expansion. \begin{conj}\label{p89conj} \begin{align*} p_8(x) &= \dfrac{1}{1680}(104594-3726282x+31805060x2 - 75882319x3)x5 \ p_9(x) &= -\dfrac{1}{180}(2235-329800x+7204710x2-48922725x3 +102061471x4)x5 \qedhere \end{align*} \end{conj}

— Asymptotic enumeration of graph factors by cumulant expansion  (2508.18731 - Isaev et al., 26 Aug 2025) in Conjecture 1 (labelled Conjecture \ref{p89conj}), Section 6, Regular graphs