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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/2508.14865 |
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| _version_ | 1866909745523195904 |
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| author | Shozi, Zekhaya B. Tacbobo, Teresa L. |
| author_facet | Shozi, Zekhaya B. Tacbobo, Teresa L. |
| contents | In this paper, we continue the study of the generator graph of a group. In 2023, Tacbobo [9] defined the generator graph of a nontrivial group to be the graph whose vertices are the elements of the group, with two vertices being adjacent if at least one of them is a generator of the group. Building on the properties established in [9], we prove that the diameter of the generator graph of a cyclic group is at most $2$. Furthermore, we present explicit formulas for some topological indices of the generator graph of a cyclic group with $n \ge 2$ elements and whose set of generators is $S$, expressed in terms of $n$ and $|S|$. Lastly, we determine the metric dimension of the generator graph of a nontrivial cyclic group as a function of its order $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the generator graph of a cyclic group Shozi, Zekhaya B. Tacbobo, Teresa L. Combinatorics In this paper, we continue the study of the generator graph of a group. In 2023, Tacbobo [9] defined the generator graph of a nontrivial group to be the graph whose vertices are the elements of the group, with two vertices being adjacent if at least one of them is a generator of the group. Building on the properties established in [9], we prove that the diameter of the generator graph of a cyclic group is at most $2$. Furthermore, we present explicit formulas for some topological indices of the generator graph of a cyclic group with $n \ge 2$ elements and whose set of generators is $S$, expressed in terms of $n$ and $|S|$. Lastly, we determine the metric dimension of the generator graph of a nontrivial cyclic group as a function of its order $n$. |
| title | On the generator graph of a cyclic group |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.14865 |