The odd-prime level bound using the last two invariant factors of the walk matrix

Let $G$ be a controllable graph, let $p$ be an odd prime factor of $\det W(G)$, and let $d_{n-1}$ and $d_n$ be the last two invariant factors of the walk matrix $W(G)$. Write $v_p$ for the $p$-adic valuation and $L(G)$ for the level of a rational regular orthogonal matrix associated with $G$. Invariant-factor level bound. One should have

$$v_p(L(G))\leq \frac{1}{2}\bigl(v_p(d_n)+v_p(d_{n-1})\bigr),$$

and, in particular,

$$v_p(L(G))\leq \frac{1}{2}v_p(\det W(G)).$$

The proposed bound would remove the technical rank assumption required in the preceding theorem and refine the determinant-based estimate by incorporating the last two invariant factors of the walk matrix.