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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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-08-01
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| Series: | Electronic Research Archive |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024222 |
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| Summary: | 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. |
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| ISSN: | 2688-1594 |