Finite groups whose coprime graphs are AT-free
Assume that $ G $ is a finite group. The coprime graph of $ G $, denoted by $ \Gamma(G) $, is an undirected graph whose vertex set is $ G $ and two distinct vertices $ x $ and $ y $ of $ \Gamma(G) $ are adjacent if and only if $ (o(x), o(y)) = 1 $, where $ o(x) $ and $ o(y) $ are the orders of $ x $...
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AIMS Press
2024-11-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024300 |
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| author | Huani Li Xuanlong Ma |
| author_facet | Huani Li Xuanlong Ma |
| author_sort | Huani Li |
| collection | DOAJ |
| description | Assume that $ G $ is a finite group. The coprime graph of $ G $, denoted by $ \Gamma(G) $, is an undirected graph whose vertex set is $ G $ and two distinct vertices $ x $ and $ y $ of $ \Gamma(G) $ are adjacent if and only if $ (o(x), o(y)) = 1 $, where $ o(x) $ and $ o(y) $ are the orders of $ x $ and $ y $, respectively. This paper gives a characterization of all finite groups with AT-free coprime graphs. This answers a question raised by Swathi and Sunitha in Forbidden subgraphs of co-prime graphs of finite groups. As applications, this paper also classifies all finite groups $ G $ such that $ \Gamma(G) $ is AT-free if $ G $ is a nilpotent group, a symmetric group, an alternating group, a direct product of two non-trivial groups, or a sporadic simple group. |
| format | Article |
| id | doaj-art-60d73fe9252e471984ba7a863e0ae0a9 |
| institution | Kabale University |
| issn | 2688-1594 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj-art-60d73fe9252e471984ba7a863e0ae0a92025-01-23T07:53:01ZengAIMS PressElectronic Research Archive2688-15942024-11-0132116443644910.3934/era.2024300Finite groups whose coprime graphs are AT-freeHuani Li0Xuanlong Ma1School of Sciences, Xi'an Technological University, Xi'an 710021, ChinaSchool of Science, Xi'an Shiyou University, Xi'an 710065, ChinaAssume that $ G $ is a finite group. The coprime graph of $ G $, denoted by $ \Gamma(G) $, is an undirected graph whose vertex set is $ G $ and two distinct vertices $ x $ and $ y $ of $ \Gamma(G) $ are adjacent if and only if $ (o(x), o(y)) = 1 $, where $ o(x) $ and $ o(y) $ are the orders of $ x $ and $ y $, respectively. This paper gives a characterization of all finite groups with AT-free coprime graphs. This answers a question raised by Swathi and Sunitha in Forbidden subgraphs of co-prime graphs of finite groups. As applications, this paper also classifies all finite groups $ G $ such that $ \Gamma(G) $ is AT-free if $ G $ is a nilpotent group, a symmetric group, an alternating group, a direct product of two non-trivial groups, or a sporadic simple group.https://www.aimspress.com/article/doi/10.3934/era.2024300coprime graphat-free graphfinite group |
| spellingShingle | Huani Li Xuanlong Ma Finite groups whose coprime graphs are AT-free Electronic Research Archive coprime graph at-free graph finite group |
| title | Finite groups whose coprime graphs are AT-free |
| title_full | Finite groups whose coprime graphs are AT-free |
| title_fullStr | Finite groups whose coprime graphs are AT-free |
| title_full_unstemmed | Finite groups whose coprime graphs are AT-free |
| title_short | Finite groups whose coprime graphs are AT-free |
| title_sort | finite groups whose coprime graphs are at free |
| topic | coprime graph at-free graph finite group |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024300 |
| work_keys_str_mv | AT huanili finitegroupswhosecoprimegraphsareatfree AT xuanlongma finitegroupswhosecoprimegraphsareatfree |