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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2020, Volume 17, Pages 1270–1279
DOI: https://doi.org/10.33048/semi.2020.17.093
(Mi semr1287)
 

Discrete mathematics and mathematical cybernetics

The nonexistence small $Q$-polynomial graphs of type (III)

A. A. Makhneva, M. M. Isakovab, A. A. Tokbaevab

a N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaya str., Yekaterinburg, 620990, Russia
b Kabardino-Balkarian State University named after H.M. Berbekov, 175, Chernyshevsky str., Nalchik, 360004, Russia
References:
Abstract: I.N. Belousov, A.A. Makhnev and M.S. Nirova found the description of $Q$-polynomial distance-regular graphs $\Gamma$ of diameter 3 such that $\Gamma_2$ and $\Gamma_3$ are strongly regular. Such graph has intersection array $\{t(c_2+1)+a_3,tc_2,a_3+1;1,c_2,t(c_2+1)\}$ and $(c_2+1)=a_3(a_3+1)/(t^2-a_3-1)$. $Q$-polynomial graph $\Gamma$ is the graph of type (I), if $a_3$ is devided by $c_2+1$, graph of type (II), if $a_3+1$ is devided by $c_2+1$, graph of type (III), if $a_3$ and $a_3+1$ does not devided by $c_2+1$.
In this paper it is proved that graph of type (III) with $t\le 6$ has intersection array $\{14,10,3;1,5,12\}$, $\{69,56,10;1,14,60\}$, $\{74,54,15;1, 9,60\}$, $\{87,66,16;1,11,72\}$, $\{119,100,15;1,20,105\}$ or $\{188,162,21;1, 27,168\}$.
Further it is proved that graphs of type (III) with intersection array $\{14,10,3;1,5,12\}$, $\{87,66,16;1,11,72\}$ and $\{188,162,21;1,27,168\}$ do not exist.
Keywords: distance-regular graph, $Q$-polynomial graph, triple intersection numbers.
Funding agency Grant number
Ministry of Education and Science of the Russian Federation 02.A03.21.0006
Received July 19, 2020, published September 7, 2020
Bibliographic databases:
Document Type: Article
UDC: 519.17
MSC: 05C25
Language: Russian
Citation: A. A. Makhnev, M. M. Isakova, A. A. Tokbaeva, “The nonexistence small $Q$-polynomial graphs of type (III)”, Sib. Èlektron. Mat. Izv., 17 (2020), 1270–1279
Citation in format AMSBIB
\Bibitem{MakIsaTok20}
\by A.~A.~Makhnev, M.~M.~Isakova, A.~A.~Tokbaeva
\paper The nonexistence small $Q$-polynomial graphs of type (III)
\jour Sib. \`Elektron. Mat. Izv.
\yr 2020
\vol 17
\pages 1270--1279
\mathnet{http://mi.mathnet.ru/semr1287}
\crossref{https://doi.org/10.33048/semi.2020.17.093}
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