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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Volume 24, Number 2, Pages 215–228
DOI: https://doi.org/10.21538/0134-4889-2018-24-2-215-228
(Mi timm1536)
 

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

On distance-regular graphs with $\theta_2=-1$

M. S. Nirova

Kabardino-Balkar State University, Nal'chik
Full-text PDF (244 kB) Citations (1)
References:
Abstract: Let a distance-regular graph $\Gamma$ of diameter 3 have eigenvalue $\theta_2=-1$. Then $\Delta=\bar\Gamma_3$ is a pseudo-geometric graph for $pG_{c_3}(k,b_1/c_2)$ containing $v$ Delsarte cliques $u^\bot$ of order $k+1$. In the case $a_1=0$ we have a partition of the subgraph $\Delta(u)$ by cliques $w^\bot-\{u\}$, where $w\in \Gamma(u)$. If there exists a strongly regular graph with parameters (176,49,12,14) in which neighborhoods of vertices are $7\times 7$-lattices, then there exists a distance-regular graph with intersection array $\{7,6,6;1,1,2\}$. If $\Delta$ contains an $n$-coclique $\{u,u_2,\dots ,u_n\}$, then there are $k_3-(n-1)(a_3+1)$ vertices in $\Gamma_3(u)-\cup_{i=2}^n \Gamma(u_i)$, which yields a new upper bound for the order of a clique in $\Gamma_3$. Moreover, it is proved that distance-regular graphs with intersection arrays $\{44,35,3;1,5,42\}$ and $\{27,20,7;1,4,21\}$ do not exist.
Keywords: distance-regular graph, eigenvalue, strongly regular graph.
Funding agency Grant number
Russian Science Foundation 18-11-00067
Received: 25.12.2017
Bibliographic databases:
Document Type: Article
UDC: 519.17
MSC: 05C25
Language: Russian
Citation: M. S. Nirova, “On distance-regular graphs with $\theta_2=-1$”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 2, 2018, 215–228
Citation in format AMSBIB
\Bibitem{Nir18}
\by M.~S.~Nirova
\paper On distance-regular graphs with $\theta_2=-1$
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2018
\vol 24
\issue 2
\pages 215--228
\mathnet{http://mi.mathnet.ru/timm1536}
\crossref{https://doi.org/10.21538/0134-4889-2018-24-2-215-228}
\elib{https://elibrary.ru/item.asp?id=35060691}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Trudy Instituta Matematiki i Mekhaniki UrO RAN
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    Full-text PDF :48
    References:29
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