A new criterion for oriented graphs to be determined by their generalized skew spectrum

Abstract: Spectral characterizations of graphs is an important topic in spectral graph theory which has been studied extensively by researchers in recent years. The study of oriented graphs, however, has received less attention so far. In Qiu et al.~\cite{QWW} (Linear Algebra Appl. 622 (2021) 316-332), the authors gave an arithmetic criterion for an oriented graph to be determined by its \emph{generalized skew spectrum} (DGSS for short). More precisely, let $\Sigma$ be an $n$-vertex oriented graph with skew adjacency matrix $S$ and $W(\Sigma)=[e,Se,\ldots,S^{n-1}e]$ be the \emph{walk-matrix} of $\Sigma$, where $e$ is the all-one vector. A theorem of Qiu et al.~\cite{QWW} shows that a self-converse oriented graph $\Sigma$ is DGSS, provided that the Smith normal form of $W(\Sigma)$ is ${\rm diag}(1,\ldots,1,2,\ldots,2,2d)$, where $d$ is an odd and square-free integer and the number of $1$'s appeared in the diagonal is precisely $\lceil \frac{n}{2}\rceil$. In this paper, we show that the above square-freeness assumptions on $d$ can actually be removed, which significantly improves upon the above theorem. Our new ingredient is a key intermediate result, which is of independent interest: for a self-converse oriented graphs $\Sigma$ and an odd prime $p$, if the rank of $W(\Sigma)$ is $n-1$ over $\mathbb{F}_p$, then the kernel of $W(\Sigma)^{\rm T}$ over $\mathbb{F}_p$ is \emph{anisotropic}, i.e., $v^{\rm T}v\neq 0$ for any $0\ne v\in{{\rm ker}\,W(\Sigma)^{\rm T}}$ over $\mathbb{F}_p$.