SHILLA GRAPHS WITH \(b=5\) AND \(b=6\)

Alexander A. Makhnev     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108; Ural Federal University, 19 Mira Str., Ekaterinburg, 620002, Russian Federation)
Ivan N. Belousov     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108; Ural Federal University, 19 Mira Str., Ekaterinburg, 620002, Russian Federation)

Abstract


A \(Q\)-polynomial Shilla graph with \(b = 5\) has intersection arrays \(\{105t,4(21t+1),16(t+1); 1,4 (t+1),84t\}\), \(t\in\{3,4,19\}\). The paper proves that distance-regular graphs with these intersection arrays do not exist. Moreover, feasible intersection arrays of \(Q\)-polynomial Shilla graphs with \(b = 6\) are found.


Keywords


Shilla graph, Distance-regular graph, Q-polynomial graph.

Full Text:

PDF

References


  1. Belousov I.N. Shilla distance-regular graphs with \(b_2 = sc_2\). Trudy Inst. Mat. i Mekh. UrO RAN, 2018. Vol. 24, No. 3. P. 16–26. (in Russian) DOI: 10.21538/0134-4889-2018-24-3-16-26
  2. Brouwer A.E., Cohen A.M., Neumaier A. Distance-Regular Graphs. Berlin, Heidelberg: Springer-Verlag, 1989. 495 p. DOI: 10.1007/978-3-642-74341-2
  3. Coolsaet K., Jurišić A. Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs. J. Combin. Theory Ser. A, 2008. Vol. 115, No. 6. P. 1086–1095. DOI: 10.1016/j.jcta.2007.12.001
  4. Gavrilyuk A.L., Koolen J.H. A characterization of the graphs of bilinear \(d\times d\)-forms over \(\mathbb{F}_2\). Combinatorica, 2019. Vol. 39, No. 2. P. 289–321. DOI: 10.1007/s00493-017-3573-4
  5. Jurišić A., Vidali J. Extremal 1-codes in distance-regular graphs of diameter 3. Des. Codes Cryptogr., 2012. Vol. 65, No. 1. P. 29–47. DOI: 10.1007/s10623-012-9651-0
  6. Koolen J.H., Park J. Shilla distance-regular graphs. European J. Combin., 2010. Vol. 31, No. 8. P. 2064–2073. DOI: 10.1016/j.ejc.2010.05.012
  7. Makhnev A.A., Belousov I.N. To the theory of Shilla graphs with \(b_2 = c_2\). Sib. Electr. Math. Reports, 2017. Vol. 14. P. 1135–1146. (in Russian) DOI: 10.17377/semi.2017.14.097




DOI: http://dx.doi.org/10.15826/umj.2021.2.004

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.