The diameter and dominating sets of the difference graph of a nilpotent group
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912801502527488 |
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| author | Ma, Xuanlong Zahirović, Samir Žigerović, Katarina |
| author_facet | Ma, Xuanlong Zahirović, Samir Žigerović, Katarina |
| contents | Given a finite group $G$, the difference graph of $G$, denoted by $\mathcal{D}(G)$, is the difference of the enhanced power graph of $G$ and the power graph of $G$, with all isolated vertices removed. This paper mainly studies the dominating sets of the difference graph of a finite group. In particular, we prove that the diameter of the difference graph of a nilpotent group has an upper bound of $4$. Furthermore, we generalize and refine the result by Biswas et al. by classifying all nilpotent groups whose difference graph has diameter $k$, for each $k\le 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01133 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The diameter and dominating sets of the difference graph of a nilpotent group Ma, Xuanlong Zahirović, Samir Žigerović, Katarina Group Theory Combinatorics 05C25 Given a finite group $G$, the difference graph of $G$, denoted by $\mathcal{D}(G)$, is the difference of the enhanced power graph of $G$ and the power graph of $G$, with all isolated vertices removed. This paper mainly studies the dominating sets of the difference graph of a finite group. In particular, we prove that the diameter of the difference graph of a nilpotent group has an upper bound of $4$. Furthermore, we generalize and refine the result by Biswas et al. by classifying all nilpotent groups whose difference graph has diameter $k$, for each $k\le 4$. |
| title | The diameter and dominating sets of the difference graph of a nilpotent group |
| topic | Group Theory Combinatorics 05C25 |
| url | https://arxiv.org/abs/2601.01133 |