Characterizing finite groups whose enhanced power graphs have universal vertices
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929238218637312 |
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| author | Costanzo, David G. Lewis, Mark L. Schmidt, Stefano Tsegaye, Eyob Udell, Gabe |
| author_facet | Costanzo, David G. Lewis, Mark L. Schmidt, Stefano Tsegaye, Eyob Udell, Gabe |
| contents | Let $G$ be a finite group and construct a graph $Δ(G)$ by taking $G\setminus\{1\}$ as the vertex set of $Δ(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set consisting of the universal vertices of $Δ(G)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient conditon for $K(G)$ to be nontrivial. We also develop a connection between $Δ(G)$ and $K(G)$ when $|G|$ is divisible by two distinct primes and the diameter of $Δ(G)$ is $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06157 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizing finite groups whose enhanced power graphs have universal vertices Costanzo, David G. Lewis, Mark L. Schmidt, Stefano Tsegaye, Eyob Udell, Gabe Group Theory Primary 20D25, Secondary 05C25 Let $G$ be a finite group and construct a graph $Δ(G)$ by taking $G\setminus\{1\}$ as the vertex set of $Δ(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set consisting of the universal vertices of $Δ(G)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient conditon for $K(G)$ to be nontrivial. We also develop a connection between $Δ(G)$ and $K(G)$ when $|G|$ is divisible by two distinct primes and the diameter of $Δ(G)$ is $2$. |
| title | Characterizing finite groups whose enhanced power graphs have universal vertices |
| topic | Group Theory Primary 20D25, Secondary 05C25 |
| url | https://arxiv.org/abs/2402.06157 |