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Vladikavkaz. Mat. Zh., 2019, Volume 21, Number 2, Pages 27–37 (Mi vmj691)  

This article is cited in 1 scientific paper (total in 1 paper)

On a distance-regular graph with an intersection array $\{35,28,6;1,2,30\}$

A. A. Makhnevab, A. A. Tokbaevac

a N. N. Krasovskii Institute of Mathematics and Mechanics, 16 S. Kovalevskaja St., Ekaterinburg 620990, Russia
b Ural Federal University, 19 Mira St., 620002 Ekaterinburg, Russia
c Kh. M. Berbekov Kabardino-Balkarian State University, 173 Chernyshevsky St., Nalchik 360004, Russia

Abstract: It is proved that for a distance-regular graph $\Gamma$ of diameter $3$ with eigenvalue $\theta_2=-1$ the complement graph of $\Gamma_3$ is pseudo-geometric for $pG_{c_3}(k,b_1/c_2 )$. Bang and Koolen investigated distance-regular graphs with intersection arrays ${(t+1)s,ts, (s+1-\psi); 1,2,(t+1)\psi}$. If $t=4$, $s=7$, $\psi=6$ then we have array ${35,28,6;1,2,30}$. Distance-regular graph $\Gamma$ with intersection array $\{35,28,6; 1,2,30\}$ has spectrum of $35^1$, $9^{168}$, $-1^{182}$, $-5^{273}$, $v=1+35+490+98=624$ vertices and $\overline{\Gamma}_3$ is a pseudogeometric graph for $pG_{30}(35,14)$. Due to the border of Delsarte, the order of clicks in $\Gamma$ is not more than $8$. It is also proved that either a neighborhood of any vertex in $\Gamma$ is the union of an isolated $7$-click, or the neighborhood of any vertex in $\Gamma$ does not contain a $7$-click and is a connected graph. The structure of the group $G$ of automorphisms of a graph $\Gamma$ with an intersection array $\{35,28,6; 1,2,30\}$ has been studied. In particular, $\pi(G)\subseteq\{2,3,5,7,13\}$ and the edge symmetric graph $\Gamma$ has a solvable group automorphisms.

Key words: distance-regular graph, Delsarte clique, geometric graph.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 02.A03.21.0006


DOI: https://doi.org/10.23671/VNC.2019.2.32115

Full text: PDF file (293 kB)
References: PDF file   HTML file

UDC: 519.17
MSC: 20D45
Received: 19.02.2019

Citation: A. A. Makhnev, A. A. Tokbaeva, “On a distance-regular graph with an intersection array $\{35,28,6;1,2,30\}$”, Vladikavkaz. Mat. Zh., 21:2 (2019), 27–37

Citation in format AMSBIB
\Bibitem{MakTok19}
\by A.~A.~Makhnev, A.~A.~Tokbaeva
\paper On a distance-regular graph with an intersection array $\{35,28,6;1,2,30\}$
\jour Vladikavkaz. Mat. Zh.
\yr 2019
\vol 21
\issue 2
\pages 27--37
\mathnet{http://mi.mathnet.ru/vmj691}
\crossref{https://doi.org/10.23671/VNC.2019.2.32115}
\elib{https://elibrary.ru/item.asp?id=39112802}


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    This publication is cited in the following articles:
    1. A. A. Makhnev, M. S. Nirova, “O distantsionno regulyarnykh grafakh s $c_2=2$”, Diskret. matem., 32:1 (2020), 74–80  mathnet  crossref  mathscinet
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