Non-Isomorphic Groups with Isomorphic Power and Commuting Graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917681930698752 |
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| author | Surbhi Venkataraman, Geetha |
| author_facet | Surbhi Venkataraman, Geetha |
| contents | The power graph of a group $G$ is a graph with vertex set $G$, in which two vertices are adjacent if one is some power of the other. In the commuting graph, with $G$ as the vertex set, two vertices are joined by an edge if they commute in $G$. The enhanced power graph of a group $G$ is a graph with vertex set $G$ and an edge joining two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. In this paper, we answer a question posed by P. J. Cameron, namely, if there exist groups $G$ and $H$ such that the power graph of $G$ is isomorphic to the commuting graph of $H$. We show that the answer is yes if $G$ is the generalised quaternion group and $H$ is the dihedral group. We also show that the enhanced power graph of the dicyclic group is isomorphic to the commuting graph of the dihedral group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_00710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Isomorphic Groups with Isomorphic Power and Commuting Graphs Surbhi Venkataraman, Geetha Group Theory Combinatorics 20D60 The power graph of a group $G$ is a graph with vertex set $G$, in which two vertices are adjacent if one is some power of the other. In the commuting graph, with $G$ as the vertex set, two vertices are joined by an edge if they commute in $G$. The enhanced power graph of a group $G$ is a graph with vertex set $G$ and an edge joining two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. In this paper, we answer a question posed by P. J. Cameron, namely, if there exist groups $G$ and $H$ such that the power graph of $G$ is isomorphic to the commuting graph of $H$. We show that the answer is yes if $G$ is the generalised quaternion group and $H$ is the dihedral group. We also show that the enhanced power graph of the dicyclic group is isomorphic to the commuting graph of the dihedral group. |
| title | Non-Isomorphic Groups with Isomorphic Power and Commuting Graphs |
| topic | Group Theory Combinatorics 20D60 |
| url | https://arxiv.org/abs/2406.00710 |