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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.07784 |
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| _version_ | 1866915235970940928 |
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| author | Lu, Yong Shen, Qi |
| author_facet | Lu, Yong Shen, Qi |
| contents | Let $\widetilde{G}=(G,U(\mathbb{Q}),φ)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph), where $G$ is the underlying graph of $\widetilde{G}$, $U(\mathbb{Q})=\{q\in \mathbb{Q}: |q|=1\}$ and $φ:\overrightarrow{E}\rightarrow U(\mathbb{Q})$ is the gain function such that $φ(e_{ij})=φ(e_{ji})^{-1}=\overline{φ(e_{ji})}$ for any adjacent vertices $v_{i}$ and $v_{j}$. Let $A(\widetilde{G})$ be the adjacency matrix of $\widetilde{G}$ and let $r(\widetilde{G})$ be the row left rank of $\widetilde{G}$. In this paper, we prove some lower bounds on the row left rank of $U(\mathbb{Q})$-gain graphs in terms of pendant vertices. All corresponding extremal graphs are characterized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07784 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The row left rank of quaternion unit gain graphs in terms of pendant vertices Lu, Yong Shen, Qi Combinatorics Let $\widetilde{G}=(G,U(\mathbb{Q}),φ)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph), where $G$ is the underlying graph of $\widetilde{G}$, $U(\mathbb{Q})=\{q\in \mathbb{Q}: |q|=1\}$ and $φ:\overrightarrow{E}\rightarrow U(\mathbb{Q})$ is the gain function such that $φ(e_{ij})=φ(e_{ji})^{-1}=\overline{φ(e_{ji})}$ for any adjacent vertices $v_{i}$ and $v_{j}$. Let $A(\widetilde{G})$ be the adjacency matrix of $\widetilde{G}$ and let $r(\widetilde{G})$ be the row left rank of $\widetilde{G}$. In this paper, we prove some lower bounds on the row left rank of $U(\mathbb{Q})$-gain graphs in terms of pendant vertices. All corresponding extremal graphs are characterized. |
| title | The row left rank of quaternion unit gain graphs in terms of pendant vertices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.07784 |