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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.18079 |
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| _version_ | 1866909593893863424 |
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| author | Lin, Limeng Wang, Wei Zhang, Hao |
| author_facet | Lin, Limeng Wang, Wei Zhang, Hao |
| contents | 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 ($\mathrm{WDGSS})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18079 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Upper Bound on the Number of Generalized Cospectral Mates of Oriented Graphs Lin, Limeng Wang, Wei Zhang, Hao Combinatorics 05C50 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 ($\mathrm{WDGSS})$. |
| title | An Upper Bound on the Number of Generalized Cospectral Mates of Oriented Graphs |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2504.18079 |