Classification of Arc-Transitive Elementary Abelian Covers of the <i>C</i>13 Graph
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> be a graph and <inline-formula><math xmlns="http://www.w3.org/1998/Math/Ma...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/5/1066 |
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| author | Qianru Xiao Aysha Khan Narges Mehdipoor Ali Asghar Talebi |
| author_facet | Qianru Xiao Aysha Khan Narges Mehdipoor Ali Asghar Talebi |
| author_sort | Qianru Xiao |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> be a graph and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>⩽</mo><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow></semantics></math></inline-formula>. A graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> can be called <i>G</i>-arc-transitive (GAT) if <i>G</i> acts transitively on its arc set. A regular covering projection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>:</mo><mover><mo>Γ</mo><mo>¯</mo></mover><mo>→</mo><mo>Γ</mo></mrow></semantics></math></inline-formula> is arc-transitive (AT) if an AT subgroup of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow></semantics></math></inline-formula> lifts under <i>p</i>. In this study, by applying a number of concepts in linear algebra such as invariant subspaces (IVs) of matrix groups (MGs), we discuss regular AT elementary abelian covers (R-AT-EA-covers) of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mn>13</mn></mrow></semantics></math></inline-formula> graph. |
| first_indexed | 2024-03-10T01:43:02Z |
| format | Article |
| id | doaj.art-2492b45acf5142e1a3df75e58420cc25 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T01:43:02Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-2492b45acf5142e1a3df75e58420cc252023-11-23T13:20:55ZengMDPI AGSymmetry2073-89942022-05-01145106610.3390/sym14051066Classification of Arc-Transitive Elementary Abelian Covers of the <i>C</i>13 GraphQianru Xiao0Aysha Khan1Narges Mehdipoor2Ali Asghar Talebi3Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, ChinaDepartment of Mathematics, Prince Sattam Bin Abdulaziz University, Al-Kharj 11991, Saudi ArabiaDepartment of Mathematics, University of Mazandaran, Babolsar 4741613534, IranDepartment of Mathematics, University of Mazandaran, Babolsar 4741613534, IranLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> be a graph and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>⩽</mo><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow></semantics></math></inline-formula>. A graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> can be called <i>G</i>-arc-transitive (GAT) if <i>G</i> acts transitively on its arc set. A regular covering projection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>:</mo><mover><mo>Γ</mo><mo>¯</mo></mover><mo>→</mo><mo>Γ</mo></mrow></semantics></math></inline-formula> is arc-transitive (AT) if an AT subgroup of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow></semantics></math></inline-formula> lifts under <i>p</i>. In this study, by applying a number of concepts in linear algebra such as invariant subspaces (IVs) of matrix groups (MGs), we discuss regular AT elementary abelian covers (R-AT-EA-covers) of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mn>13</mn></mrow></semantics></math></inline-formula> graph.https://www.mdpi.com/2073-8994/14/5/1066MGsIVshomology group<i>C</i>13 graphAT graphsregular covering |
| spellingShingle | Qianru Xiao Aysha Khan Narges Mehdipoor Ali Asghar Talebi Classification of Arc-Transitive Elementary Abelian Covers of the <i>C</i>13 Graph Symmetry MGs IVs homology group <i>C</i>13 graph AT graphs regular covering |
| title | Classification of Arc-Transitive Elementary Abelian Covers of the <i>C</i>13 Graph |
| title_full | Classification of Arc-Transitive Elementary Abelian Covers of the <i>C</i>13 Graph |
| title_fullStr | Classification of Arc-Transitive Elementary Abelian Covers of the <i>C</i>13 Graph |
| title_full_unstemmed | Classification of Arc-Transitive Elementary Abelian Covers of the <i>C</i>13 Graph |
| title_short | Classification of Arc-Transitive Elementary Abelian Covers of the <i>C</i>13 Graph |
| title_sort | classification of arc transitive elementary abelian covers of the i c i 13 graph |
| topic | MGs IVs homology group <i>C</i>13 graph AT graphs regular covering |
| url | https://www.mdpi.com/2073-8994/14/5/1066 |
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