ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\)
For a distance-regular graph \(\Gamma\) of diameter 3, the graph \(\Gamma_i\) can be strongly regular for \(i=2\) or 3. J.Kulen and co-authors found the parameters of a strongly regular graph \(\Gamma_2\) given the intersection array of the graph \(\Gamma\) (independently, the parameters were found...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2022-12-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/463 |
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| author | Alexander A. Makhnev Ivan N. Belousov Konstantin S. Efimov |
| author_facet | Alexander A. Makhnev Ivan N. Belousov Konstantin S. Efimov |
| author_sort | Alexander A. Makhnev |
| collection | DOAJ |
| description | For a distance-regular graph \(\Gamma\) of diameter 3, the graph \(\Gamma_i\) can be strongly regular for \(i=2\) or 3. J.Kulen and co-authors found the parameters of a strongly regular graph \(\Gamma_2\) given the intersection array of the graph \(\Gamma\) (independently, the parameters were found by A.A. Makhnev and D.V.Paduchikh). In this case, \(\Gamma\) has an eigenvalue \(a_2-c_3\). In this paper, we study graphs \(\Gamma\) with strongly regular graph \(\Gamma_2\) and eigenvalue \(\theta=1\). In particular, we prove that, for a \(Q\)-polynomial graph from a series of graphs with intersection arrays \(\{2c_3+a_1+1,2c_3,c_3+a_1-c_2;1,c_2,c_3\}\), the equality \(c_3=4 (t^2+t)/(4t+4-c_2^2)\) holds. Moreover, for \(t\le 100000\), there is a unique feasible intersection array \(\{9,6,3;1,2,3\}\) corresponding to the Hamming (or Doob) graph \(H(3,4)\). In addition, we found parametrizations of intersection arrays of graphs with \(\theta_2=1\) and \(\theta_3=a_2-c_3\). |
| format | Article |
| id | doaj-art-9af5fec77d8e4174b7e0e9d8cdede0d5 |
| institution | Kabale University |
| issn | 2414-3952 |
| language | English |
| publishDate | 2022-12-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-9af5fec77d8e4174b7e0e9d8cdede0d52025-08-20T03:33:46ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522022-12-018210.15826/umj.2022.2.010164ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\)Alexander A. Makhnev0Ivan N. Belousov1Konstantin S. Efimov2Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation; Ural Federal University, 19 Mira str., Ekaterinburg, 620002Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation; Ural Federal University, 19 Mira str., Ekaterinburg, 620002Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation; Ural Federal University, 19 Mira str., Ekaterinburg, 620002For a distance-regular graph \(\Gamma\) of diameter 3, the graph \(\Gamma_i\) can be strongly regular for \(i=2\) or 3. J.Kulen and co-authors found the parameters of a strongly regular graph \(\Gamma_2\) given the intersection array of the graph \(\Gamma\) (independently, the parameters were found by A.A. Makhnev and D.V.Paduchikh). In this case, \(\Gamma\) has an eigenvalue \(a_2-c_3\). In this paper, we study graphs \(\Gamma\) with strongly regular graph \(\Gamma_2\) and eigenvalue \(\theta=1\). In particular, we prove that, for a \(Q\)-polynomial graph from a series of graphs with intersection arrays \(\{2c_3+a_1+1,2c_3,c_3+a_1-c_2;1,c_2,c_3\}\), the equality \(c_3=4 (t^2+t)/(4t+4-c_2^2)\) holds. Moreover, for \(t\le 100000\), there is a unique feasible intersection array \(\{9,6,3;1,2,3\}\) corresponding to the Hamming (or Doob) graph \(H(3,4)\). In addition, we found parametrizations of intersection arrays of graphs with \(\theta_2=1\) and \(\theta_3=a_2-c_3\).https://umjuran.ru/index.php/umj/article/view/463strongly regular graph, distance-regular graph, intersection array |
| spellingShingle | Alexander A. Makhnev Ivan N. Belousov Konstantin S. Efimov ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\) Ural Mathematical Journal strongly regular graph, distance-regular graph, intersection array |
| title | ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\) |
| title_full | ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\) |
| title_fullStr | ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\) |
| title_full_unstemmed | ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\) |
| title_short | ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\) |
| title_sort | on distance regular graphs of diameter 3 with eigenvalue theta 1 |
| topic | strongly regular graph, distance-regular graph, intersection array |
| url | https://umjuran.ru/index.php/umj/article/view/463 |
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