The strong spectral property for some families of unicyclic graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909447640580096 |
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| author | Koljančić, Sara Oblak, Polona |
| author_facet | Koljančić, Sara Oblak, Polona |
| contents | To find all the possible spectra of all real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of a given graph $G$, the Strong Spectral Property turned out to be of crucial importance. In particular, we investigate the set ${\mathcal G}^{\text{SSP}}$ of all simple graphs $G$ with the property that each symmetric matrix of the pattern of $G$ has the Strong Spectral Property. In this paper, we completely characterize unicyclic graphs of girth three in ${\mathcal G}^{\text{SSP}}$. We prove that any tadpole graph of girth at most five is in ${\mathcal G}^{\text{SSP}}$ and we show that the same is not valid for girth six tadpole graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The strong spectral property for some families of unicyclic graphs Koljančić, Sara Oblak, Polona Combinatorics 05C50, 05C38, 15A18, 15B57, 65F18 To find all the possible spectra of all real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of a given graph $G$, the Strong Spectral Property turned out to be of crucial importance. In particular, we investigate the set ${\mathcal G}^{\text{SSP}}$ of all simple graphs $G$ with the property that each symmetric matrix of the pattern of $G$ has the Strong Spectral Property. In this paper, we completely characterize unicyclic graphs of girth three in ${\mathcal G}^{\text{SSP}}$. We prove that any tadpole graph of girth at most five is in ${\mathcal G}^{\text{SSP}}$ and we show that the same is not valid for girth six tadpole graphs. |
| title | The strong spectral property for some families of unicyclic graphs |
| topic | Combinatorics 05C50, 05C38, 15A18, 15B57, 65F18 |
| url | https://arxiv.org/abs/2501.01719 |