Almost all graphs have no cospectral mate with fixed level
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911141454675968 |
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| author | Wang, Wei Zhao, Da |
| author_facet | Wang, Wei Zhao, Da |
| contents | Haemers conjectures that almost all graphs are determined by their spectra. Suppose $G \sim \mathcal{G}(n, p)$ is a random graph with each edge chosen independently with probability $p$ with $0 < p < 1$. Then $$\Pr(G \text{ is not controllable}) + \sum_{\ell = 2}^{n^{n^2}} \Pr(G \text{ has a generalized copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ implies that almost all graphs are determined by their generalized spectra. It is known that almost all graphs are controllable. We show that almost all graphs have no cospectral mate with fixed level $\ell$, namely $$\Pr(G \text{ has a copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ for every $\ell \geq 2$. Consequently, $$\Pr(G \text{ has a generalized copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ for every $\ell \geq 2$. The result can also be interpreted in the framework of random integral matrices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_05781 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost all graphs have no cospectral mate with fixed level Wang, Wei Zhao, Da Combinatorics 05C50, 05C80, 60B20, 60C05 Haemers conjectures that almost all graphs are determined by their spectra. Suppose $G \sim \mathcal{G}(n, p)$ is a random graph with each edge chosen independently with probability $p$ with $0 < p < 1$. Then $$\Pr(G \text{ is not controllable}) + \sum_{\ell = 2}^{n^{n^2}} \Pr(G \text{ has a generalized copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ implies that almost all graphs are determined by their generalized spectra. It is known that almost all graphs are controllable. We show that almost all graphs have no cospectral mate with fixed level $\ell$, namely $$\Pr(G \text{ has a copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ for every $\ell \geq 2$. Consequently, $$\Pr(G \text{ has a generalized copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ for every $\ell \geq 2$. The result can also be interpreted in the framework of random integral matrices. |
| title | Almost all graphs have no cospectral mate with fixed level |
| topic | Combinatorics 05C50, 05C80, 60B20, 60C05 |
| url | https://arxiv.org/abs/2509.05781 |