A New Sufficient Condition for Oriented Graphs Determined by Their Generalized Skew Spectra

Abstract

Characterizing graphs uniquely determined by their spectra (DS) is a core open problem in spectral graph theory. While this problem has been extensively investigated for simple undirected graphs, it remains relatively underexplored for oriented graphs. For a simple undirected graph $G$ equipped with an orientation $\sigma$, the corresponding oriented graph $\Sigma =(G,\sigma )$ is the digraph obtained by orienting each edge of $G$ according to $\sigma$. An oriented graph $\Sigma$ is said to be \emph{determined by its generalized skew spectrum} (DGSS) if every oriented graph sharing the same generalized skew spectrum is isomorphic to $\Sigma$. This paper develops a new sufficient criterion for recognizing DGSS controllable oriented graphs, which applies to a much broader family of graphs than previously known results. Let $S$ be the skew-adjacency matrix of $\Sigma$, $W(\Sigma )=[e,Se,\dots ,S^{n-1}e]$, and $d_n$ the last invariant factor of $W(\Sigma )$. For each odd prime $p$, we define the polynomial ${\Phi }_p(\Sigma ;x)=\gcd (\chi (S;x),\chi (S+J;x))$ over the finite field ${\mathbb{F}}_p$, which is invariant under generalized skew cospectrality. By analyzing the square-free part of ${\Phi }_p(\Sigma ;x)$ and the associated $p$-main polynomial, we establish a DGSS sufficient condition under the square-free assumption on $d_n$. The proposed criterion allows higher $p$-nullity and recovers the square-free determinant criterion of Qiu, Wang and Wang~(2019) as a special case. We further provide illustrative examples to verify the wider applicability of our new condition and to highlight the role of the compatibility constraints on the irreducible factors of ${\Phi }_p(\Sigma ;x)$.