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

Source 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 $σ$, the corresponding oriented graph $Σ=(G,σ)$ is the digraph obtained by orienting each edge of $G$ according to $σ$. An oriented graph $Σ$ 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 $Σ$. 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 $Σ$, $W(Σ)=[e,Se,\ldots,S^{n-1}e]$, and $d_n$ the last invariant factor of $W(Σ)$. For each odd prime $p$, we define the polynomial $Φ_p(Σ;x)=\gcd(χ(S;x),χ(S+J;x))$ over the finite field $\mathbb{F}_p$, which is invariant under generalized skew cospectrality. By analyzing the square-free part of $Φ_p(Σ;x)$ 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 $Φ_p(Σ;x)$.