The walk-matrix criterion for generalized spectral determination

Let $G$ be a self-converse mixed graph of order $n$, with walk-matrix

$$W(G)=[e,Ae,A^2e,\ldots,A^{n-1}e].$$

Here $\mathcal{G}_n^{sc}$ denotes the self-converse mixed graphs of order $n$, and $G$ is DGS when it is determined by its generalized Hermitian spectrum. Walk-matrix DGS conjecture. If $G\in\mathcal{G}_n^{sc}$ and

$$\frac{|\det W(G)|}{2^{\lfloor n/2\rfloor}}$$

is odd and square-free, then $G$ is DGS. This conjectural criterion generalizes a known divisibility phenomenon for ordinary graph walk-matrices to self-converse mixed graphs; the paper presents it as a proposed generalization and develops equivalent formulations.