On the levels of rational regular orthogonal matrices for generalized cospectral graphs

For an $n$-vertex graph $G$ with adjacency matrix $A$, the walk matrix $W(G)$ of $G$ is the matrix $[e,Ae,\ldots,A^{n-1}e]$, where $e$ is the all-ones vector. Suppose that $W(G)$ is nonsingular and $p$ is an odd prime such that $W(G)$ has rank $n-1$ over the finite field $\mathbb{Z}/p\mathbb{Z}$. Let $H$ be a graph that is generalized cospectral with $G$, and $Q$ be the corresponding rational regular orthogonal matrix satisfying $Q^\mathsf{T} A(G) Q=A(H)$. We prove that \begin{equation*} v_p(\ell(Q))\le \frac{1}{2}v_p (\det W(G)) \end{equation*} where $\ell(Q)$ is the minimum positive integer $k$ such that $kQ$ is an integral matrix, and $v_p(m)$ is the maximum nonnegative integer $s$ such that $p^s$ divides $m$. This significantly improves upon a recent result of Qiu et al. [Discrete Math. 346 (2023) 113177] stating that $v_p(\ell(Q))\le v_p (\det W(G))-1.$