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Low Diameter Algebraic Graphs

Abstract : Let D be a division ring, n a positive integer, and GLn(D) the set of invertible square matrices of size n and values in D, called the general linear group. We address the intersection graph of subgroups of GLn(D) and prove that it has diameter at most 3. Two particular cases of its induced subgraphs are then investigated: by cyclic subgroups, and by almost subnormal subgroups. We prove that the latter case results in a connected graph whose diameter is sharply bounded by 2. In the former case, we completely characterise the connectivity of the induced graph with respect to D, where, in case of connectivity, we prove that it has diameter at most 7 in general, and at most 5 if D is a locally finite field of characteristic not 2 different from F3 and F9.
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Contributor : Binh-Minh Bui-Xuan Connect in order to contact the contributor
Submitted on : Wednesday, November 17, 2021 - 7:38:35 PM
Last modification on : Sunday, June 26, 2022 - 3:19:52 AM
Long-term archiving on: : Friday, February 18, 2022 - 9:00:35 PM


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Hai Bui Xuan, Binh-Minh Bui-Xuan, Le van Chua, Mai Hoang Bien. Low Diameter Algebraic Graphs. European Conference on Combinatorics, Graph Theory and Applications, Sep 2021, Barcelona, Spain. pp.465-471, ⟨10.1007/978-3-030-83823-2_73⟩. ⟨hal-03433624⟩



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