Divisibility pattern for higher-order expansion polynomials

Establish whether, for every integer j greater than 7, the polynomial p_j(x) in the asymptotic expansion for the number of d-regular graphs contains x^{\lfloor j/2\rfloor+1} as a divisor, and consequently whether the expansion with a decreasing error term extends to constant degree d.

Background

The paper derives an asymptotic expansion for the number of labelled d-regular graphs and explicitly computes the polynomials p_j(x) through p_7. The authors observe that each computed polynomial p_j(x), for 2 ≤ j ≤ 7, is divisible by x{\lfloor j/2\rfloor+1}.

They report experimental evidence that this pattern persists for larger indices and that the resulting expansion may remain valid even when the degree d is constant. However, the observed divisibility pattern is not proved, leaving both the all-j statement and its implication for constant-degree asymptotics unresolved.

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 6, Regular graphs, discussion following the explicit formulas for p_1 through p_7