An Upper Bound on the Number of Generalized Cospectral Mates of Oriented Graphs

TL;DR

This work extends spectral characterization of graphs to oriented graphs by analyzing the generalized skew-spectrum. It introduces the class $\mathcal{G}_{n}$ via the invariant $2^{-ig\lfloor n/2\big\rfloor}\det W(\Sigma)$ and links cospectrality with the converse $\Sigma^{\rm T}$ to a unique regular rational orthogonal matrix $Q_{0}$ whose level $\ell_{0}$ critically bounds the number of generalized cospectral mates. The main result shows the maximum number of such mates is $2^{t}-1$, where $t$ is the number of distinct odd prime factors of $\ell_{0}$, and demonstrates attainability with concrete examples; the paper also gives a criterion to identify oriented graphs in $\mathcal{G}_{n}$ that are WD GSS. Together, these findings advance understanding of when oriented graphs are determined by their generalized skew-spectrum and how many cospectral relatives may exist, with practical implications for identifying WD GSS examples and guiding future work on larger $t$.

Abstract

This paper examines the spectral characterizations of oriented graphs. Let $Σ$ be an $n$-vertex oriented graph with skew-adjacency matrix $S$. Previous research mainly focused on self-converse oriented graphs, proposing arithmetic conditions for these graphs to be uniquely determined by their generalized skew-spectrum ($\mathrm{DGSS}$). However, self-converse graphs are extremely rare; this paper considers a more general class of oriented graphs $\mathcal{G}_{n}$ (not limited to self-converse graphs), consisting of all $n$-vertex oriented graphs $Σ$ such that $2^{-\left \lfloor \frac{n}{2} \right \rfloor }\det W(Σ)$ is an odd and square-free integer, where $W(Σ)=[e,Se,\dots,S^{n-1}e]$ ($e$ is the all-one vector) is the skew-walk matrix of $Σ$. Given that $Σ$ is cospectral with its converse $Σ^{\rm T}$, there always exists a unique regular rational orthogonal $Q_0$ such that $Q_0^{\rm T}SQ_0=-S$. This study reveals that there exists a deep relationship between the level $\ell_0$ of $Q_0$ and the number of generalized cospectral mates of $Σ$. More precisely, we show, among others, that the maximum number of generalized cospectral mates of $Σ\in\mathcal{G}_{n}$ is at most $2^{t}-1$, where $t$ is the number of prime factors of $\ell_0$. Moreover, some numerical examples are also provided to demonstrate that the above upper bound is attainable. Finally, we also provide a criterion for the oriented graphs $Σ\in\mathcal{G}_{n}$ to be weakly determined by the generalized skew-spectrum ($\mathrm{WDGSS})$.

Figures (4)

• Figure 1: The relationship between all cospectral mates of $\Sigma$ with $\ell_0=p_1p_2$.
• Figure 2: A non-self-converse oriented graph $\Sigma$ which is $\mathrm{WDGSS}$.
• Figure 3: A non-self-converse oriented graphs $\Sigma$.
• Figure 4: A non-self-converse oriented graphs $\Sigma$

Theorems & Definitions (60)

• Theorem 1: Wang W4
• Theorem 2: ref4
• Theorem 3
• Remark 1
• Remark 2
• Theorem 4
• Remark 3
• Theorem 5
• Lemma 1: ref4,ref7
• Definition 1: ref7