On forbidden subgraphs of main supergraphs of groups
In this study, we explore the main supergraph $ \mathcal{S}(G) $ of a finite group $ G $, defined as an undirected, simple graph with a vertex set $ G $ in which two distinct vertices, $ a $ and $ b $, are adjacent in $ \mathcal{S}(G) $ if the order of one is a divisor of the order of the other. Thi...
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| Language: | English |
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AIMS Press
2024-08-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024222 |
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| author | Xiaoyan Xu Xiaohua Xu Jin Chen Shixun Lin |
| author_facet | Xiaoyan Xu Xiaohua Xu Jin Chen Shixun Lin |
| author_sort | Xiaoyan Xu |
| collection | DOAJ |
| description | In this study, we explore the main supergraph $ \mathcal{S}(G) $ of a finite group $ G $, defined as an undirected, simple graph with a vertex set $ G $ in which two distinct vertices, $ a $ and $ b $, are adjacent in $ \mathcal{S}(G) $ if the order of one is a divisor of the order of the other. This is denoted as either $ o(a)\mid o(b) $ or $ o(b)\mid o(a) $, where $ o(\cdot) $ is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups $ G $ with $ \mathcal{S}(G) $, a cograph that includes when $ G $ is a direct product of two non-trivial groups, as well as when $ G $ is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group. |
| format | Article |
| id | doaj-art-79ca19c08347414c9d43e51c26a04637 |
| institution | Kabale University |
| issn | 2688-1594 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj-art-79ca19c08347414c9d43e51c26a046372025-01-23T07:51:27ZengAIMS PressElectronic Research Archive2688-15942024-08-013284845485710.3934/era.2024222On forbidden subgraphs of main supergraphs of groupsXiaoyan Xu0Xiaohua Xu1Jin Chen2Shixun Lin3School of Science, Xi'an Shiyou University, Xi'an 710065, ChinaInformation Technology Education Center, Zhaotong University, Zhaotong 657000, ChinaSchool of Mathematics and Statistics, Zhaotong University, Zhaotong 657000, ChinaSchool of Mathematics and Statistics, Zhaotong University, Zhaotong 657000, ChinaIn this study, we explore the main supergraph $ \mathcal{S}(G) $ of a finite group $ G $, defined as an undirected, simple graph with a vertex set $ G $ in which two distinct vertices, $ a $ and $ b $, are adjacent in $ \mathcal{S}(G) $ if the order of one is a divisor of the order of the other. This is denoted as either $ o(a)\mid o(b) $ or $ o(b)\mid o(a) $, where $ o(\cdot) $ is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups $ G $ with $ \mathcal{S}(G) $, a cograph that includes when $ G $ is a direct product of two non-trivial groups, as well as when $ G $ is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group.https://www.aimspress.com/article/doi/10.3934/era.2024222main supergraphfinite groupsplit graphthreshold graphcograph |
| spellingShingle | Xiaoyan Xu Xiaohua Xu Jin Chen Shixun Lin On forbidden subgraphs of main supergraphs of groups Electronic Research Archive main supergraph finite group split graph threshold graph cograph |
| title | On forbidden subgraphs of main supergraphs of groups |
| title_full | On forbidden subgraphs of main supergraphs of groups |
| title_fullStr | On forbidden subgraphs of main supergraphs of groups |
| title_full_unstemmed | On forbidden subgraphs of main supergraphs of groups |
| title_short | On forbidden subgraphs of main supergraphs of groups |
| title_sort | on forbidden subgraphs of main supergraphs of groups |
| topic | main supergraph finite group split graph threshold graph cograph |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024222 |
| work_keys_str_mv | AT xiaoyanxu onforbiddensubgraphsofmainsupergraphsofgroups AT xiaohuaxu onforbiddensubgraphsofmainsupergraphsofgroups AT jinchen onforbiddensubgraphsofmainsupergraphsofgroups AT shixunlin onforbiddensubgraphsofmainsupergraphsofgroups |