Finite groups whose commuting graphs are line graphs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918068622458880 |
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| author | Malviy, Siddharth Kakkar, Vipul |
| author_facet | Malviy, Siddharth Kakkar, Vipul |
| contents | The commuting graph ${Γ(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all the dominant vertices of $Γ(G)$, the resulting subgraphs of $Γ(G)$ are $Γ^*(G)$ and $Γ^{**}(G)$, respectively. In this paper, we classify all the finite groups $G$ such that the graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the line graph of some graph. We also classify all the finite groups $G$ whose graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the complement of line graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04495 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite groups whose commuting graphs are line graphs Malviy, Siddharth Kakkar, Vipul Combinatorics The commuting graph ${Γ(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all the dominant vertices of $Γ(G)$, the resulting subgraphs of $Γ(G)$ are $Γ^*(G)$ and $Γ^{**}(G)$, respectively. In this paper, we classify all the finite groups $G$ such that the graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the line graph of some graph. We also classify all the finite groups $G$ whose graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the complement of line graph. |
| title | Finite groups whose commuting graphs are line graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.04495 |