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

Yiquan Chao, Wei Wang, Hao Zhang

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 $Σ$ be an $n$-vertex oriented graph with skew adjacency matrix $S$ and $W(Σ)=[e,Se,\ldots,S^{n-1}e]$ be the \emph{walk-matrix} of $Σ$, where $e$ is the all-one vector. A theorem of Qiu et al.~\cite{QWW} shows that a self-converse oriented graph $Σ$ is DGSS, provided that the Smith normal form of $W(Σ)$ 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 $Σ$ and an odd prime $p$, if the rank of $W(Σ)$ is $n-1$ over $\mathbb{F}_p$, then the kernel of $W(Σ)^{\rm T}$ over $\mathbb{F}_p$ is \emph{anisotropic}, i.e., $v^{\rm T}v\neq 0$ for any $0\ne v\in{{\rm ker}\,W(Σ)^{\rm T}}$ over $\mathbb{F}_p$.

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 be an -vertex oriented graph with skew adjacency matrix and be the \emph{walk-matrix} of , where is the all-one vector. A theorem of Qiu et al.~\cite{QWW} shows that a self-converse oriented graph is DGSS, provided that the Smith normal form of is , where is an odd and square-free integer and the number of 's appeared in the diagonal is precisely . In this paper, we show that the above square-freeness assumptions on 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 and an odd prime , if the rank of is over , then the kernel of over is \emph{anisotropic}, i.e., for any over .

Paper Structure

This paper contains 6 sections, 15 theorems, 16 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The main strategy
  4. Some lemmas
  5. Proofs of Theorem \ref{['main']} and Theorem \ref{['tournament']}
  6. Some examples

Key Result

Theorem 1

Let $G$ be graph of order $n$. If $2^{-\lfloor\frac{n}{2}\rfloor}\det W(G)$ (which is always an integer) is odd and square-free, then $G$ is DGS.

Figures (4)

  • Figure 1: An oriented graph $\Sigma$ with skew adjacency matrix $S$.
  • Figure 2: A self-converse oriented graph $\Sigma$
  • Figure 3: A self-conversed DGSS oriented graph $\Sigma$
  • Figure 4: A self-conversed DGSS oriented graph $\Sigma$

Theorems & Definitions (30)

  • Theorem 1: Wang W4
  • Example 1
  • Definition 1
  • Example 2
  • Remark 1
  • Theorem 2: Qiu et al. QWW
  • Theorem 3
  • Definition 2
  • Theorem 4
  • Lemma 1: c.f. Wang and Xu W1
  • ...and 20 more