Divisibility pattern for the regular-graph expansion polynomials

Prove that, for every integer j greater than or equal to 2, the polynomial p_j(x) in the asymptotic expansion of the number of labelled d-regular graphs has x^{\lfloor j/2\rfloor+1} as a divisor, and thereby determine whether the expansion remains valid with a decreasing error term for constant degree d.

Background

The paper derives an asymptotic expansion for the number of labelled d-regular graphs, with coefficient polynomials p_j(x). Explicit computation through p_7 shows that p_j(x) contains x{\lfloor j/2\rfloor+1} as a factor for 2\leq j\leq 7. Numerical experiments for fixed small degrees suggest that the expansion may continue to hold beyond the theorem’s stated range, potentially including constant d, but the observed divisibility pattern has not been established in general.

References

It is interesting to note that $p_j(x)$ has $x{\lfloor j/2\rfloor+1}$ as a divisor for $2\leq j\leq 7$. If this is true for all larger~$j$, Theorem~\ref{regularthm} with a decreasing error term may hold even for constant~$d$. This is what happens experimentally, as illustrated in Figure~\ref{plot} for~$k=7$. However, we stress that we have not proved this observation.

— Asymptotic enumeration of graph factors by cumulant expansion  (2508.18731 - Isaev et al., 26 Aug 2025) in Section “Regular graphs”, immediately following Figure 1 and preceding Conjecture 1