On Some Regular Two-Graphs up to 50 Vertices
Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/2/408 |
| _version_ | 1797618079798132736 |
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| author | Marija Maksimović |
| author_facet | Marija Maksimović |
| author_sort | Marija Maksimović |
| collection | DOAJ |
| description | Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>45</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>11</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>49</mn><mo>,</mo><mn>24</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>50</mn><mo>,</mo><mn>21</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>9</mn><mo>)</mo></mrow></semantics></math></inline-formula> that have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mn>6</mn></msub></semantics></math></inline-formula> as the automorphism group and constructed regular two-graphs from SRGs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>45</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>11</mn><mo>)</mo></mrow></semantics></math></inline-formula>, SRGs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>49</mn><mo>,</mo><mn>24</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and SRGs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>50</mn><mo>,</mo><mn>21</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>9</mn><mo>)</mo></mrow></semantics></math></inline-formula> that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>45</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>11</mn><mo>)</mo></mrow></semantics></math></inline-formula> and 51 new regular two-graphs on 50 vertices leading to 398 new SRG <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>49</mn><mo>,</mo><mn>24</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T08:05:10Z |
| format | Article |
| id | doaj.art-b27249215ea848b5a47031d2b5537284 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T08:05:10Z |
| publishDate | 2023-02-01 |
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| series | Symmetry |
| spelling | doaj.art-b27249215ea848b5a47031d2b55372842023-11-16T23:32:55ZengMDPI AGSymmetry2073-89942023-02-0115240810.3390/sym15020408On Some Regular Two-Graphs up to 50 VerticesMarija Maksimović0Faculty of Mathematics, University of Rijeka, 51000 Rijeka, CroatiaRegular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>45</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>11</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>49</mn><mo>,</mo><mn>24</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>50</mn><mo>,</mo><mn>21</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>9</mn><mo>)</mo></mrow></semantics></math></inline-formula> that have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mn>6</mn></msub></semantics></math></inline-formula> as the automorphism group and constructed regular two-graphs from SRGs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>45</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>11</mn><mo>)</mo></mrow></semantics></math></inline-formula>, SRGs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>49</mn><mo>,</mo><mn>24</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and SRGs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>50</mn><mo>,</mo><mn>21</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>9</mn><mo>)</mo></mrow></semantics></math></inline-formula> that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>45</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>11</mn><mo>)</mo></mrow></semantics></math></inline-formula> and 51 new regular two-graphs on 50 vertices leading to 398 new SRG <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>49</mn><mo>,</mo><mn>24</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2073-8994/15/2/408orbit matricesgraphstwo-graphsregular two-graphsstrongly regular graphs |
| spellingShingle | Marija Maksimović On Some Regular Two-Graphs up to 50 Vertices Symmetry orbit matrices graphs two-graphs regular two-graphs strongly regular graphs |
| title | On Some Regular Two-Graphs up to 50 Vertices |
| title_full | On Some Regular Two-Graphs up to 50 Vertices |
| title_fullStr | On Some Regular Two-Graphs up to 50 Vertices |
| title_full_unstemmed | On Some Regular Two-Graphs up to 50 Vertices |
| title_short | On Some Regular Two-Graphs up to 50 Vertices |
| title_sort | on some regular two graphs up to 50 vertices |
| topic | orbit matrices graphs two-graphs regular two-graphs strongly regular graphs |
| url | https://www.mdpi.com/2073-8994/15/2/408 |
| work_keys_str_mv | AT marijamaksimovic onsomeregulartwographsupto50vertices |