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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2405.17154 |
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| _version_ | 1866913364929675264 |
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| author | Andriantiana, Eric O. D. Shozi, Zekhaya B. |
| author_facet | Andriantiana, Eric O. D. Shozi, Zekhaya B. |
| contents | Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges $E(G)$. A subset $S$ of $E(G)$ is called a $k$-nearly independent edge subsets if there are exactly $k$ pairs of elements of $S$ that share a common end. $Z_k(G)$ is the number of such subsets. This paper studies $Z_1$. Various properties of $Z_1$ are discussed. We characterise the two $n$-vertex trees with smallest $Z_1$, as well as the one with largest value. A conjecture on the $n$-vertex tree with second-largest $Z_1$ is proposed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17154 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The number of 1-nearly independent edge subsets Andriantiana, Eric O. D. Shozi, Zekhaya B. Combinatorics Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges $E(G)$. A subset $S$ of $E(G)$ is called a $k$-nearly independent edge subsets if there are exactly $k$ pairs of elements of $S$ that share a common end. $Z_k(G)$ is the number of such subsets. This paper studies $Z_1$. Various properties of $Z_1$ are discussed. We characterise the two $n$-vertex trees with smallest $Z_1$, as well as the one with largest value. A conjecture on the $n$-vertex tree with second-largest $Z_1$ is proposed. |
| title | The number of 1-nearly independent edge subsets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.17154 |