由广义斜谱确定的有向图的一个新充分条件

中文摘要 AI 辅助

刻画由其谱唯一确定的图（DS图）是谱图论中的核心开放问题。尽管该问题已被广泛研究简单无向图，但对于有向图仍研究较少。对于配备定向σ的简单无向图G，对应有向图Σ=(G,σ)是通过按σ定向G的每条边得到的有向图。若每个具有相同广义斜谱的有向图都与Σ同构，则称Σ是由其广义斜谱确定的图（DGSS图）。本文提出了识别DGSS可控有向图的新充分准则，适用于比以往结果更广泛的图族。设S为Σ的斜邻接矩阵，W(Σ)=[e,Se,…,S^{n-1}e]，d_n为W(Σ)的最后不变因子。对每个奇素数p，在有限域F_p上定义多项式Φ_p(Σ;x)=gcd(χ(S;x),χ(S+J;x))，该多项式在广义斜共谱下不变。通过分析Φ_p(Σ;x)的无平方因子部分及相关p-主多项式，在d_n无平方因子的假设下建立了DGSS充分条件。该准则允许更高的p-零空间维度，且将Qiu、Wang和Wang（2019）的无平方因子行列式准则作为特例。本文还提供示例验证新条件的更广适用性，并强调Φ_p(Σ;x)不可约因子的相容性约束的作用。

英文摘要

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 $Φ_p(Σ;x)$.

发表机构

• School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
• School of Mathematics, Physics and Finance, Anhui Polytechnic University（安徽工程大学数理金融学院）
• School of Mathematics, Hunan University（湖南大学数学学院）

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