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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.15001 |
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| _version_ | 1866914955211571200 |
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| author | Reimbayev, Reimbay |
| author_facet | Reimbayev, Reimbay |
| contents | For a locally linear graph $G$, which is a graph built out of triangles, it is possible to construct another graph $G^*$ that would consist of triangles of $G$ as vertices, while sharing (or not sharing) a common vertex between a pair of triangles would define a binary relation for edges of $G^*$. In this paper we show that the spectrum of $G^*$ is uniquely defined by $G$. We will also show some structural similarities of these graphs; in particular, that the number of quadrilaterals and pentagons in both graphs are the same; that $G^*$ does not contain $K_4-e$ and $K_{1,4}$; and that $G$ can be reconstructed from $G^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15001 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Spectrum of Locally Linear Graphs Reimbayev, Reimbay Combinatorics 05C75, 05C50 For a locally linear graph $G$, which is a graph built out of triangles, it is possible to construct another graph $G^*$ that would consist of triangles of $G$ as vertices, while sharing (or not sharing) a common vertex between a pair of triangles would define a binary relation for edges of $G^*$. In this paper we show that the spectrum of $G^*$ is uniquely defined by $G$. We will also show some structural similarities of these graphs; in particular, that the number of quadrilaterals and pentagons in both graphs are the same; that $G^*$ does not contain $K_4-e$ and $K_{1,4}$; and that $G$ can be reconstructed from $G^*$. |
| title | On the Spectrum of Locally Linear Graphs |
| topic | Combinatorics 05C75, 05C50 |
| url | https://arxiv.org/abs/2409.15001 |