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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Volume 24, Number 3, Pages 133–144
DOI: https://doi.org/10.21538/0134-4889-2018-24-3-133-144
(Mi timm1557)
 

This article is cited in 2 scientific papers (total in 2 papers)

Inverse problems in distance-regular graphs theory

A. A. Makhnevab, D. V. Paduchikha

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Full-text PDF (215 kB) Citations (2)
References:
Abstract: For a distance-regular graph $\Gamma$ of diameter 3, the graph $\Gamma_i$ can be strongly regular for $i=2$ or $3$. Finding the parameters of $\Gamma_i$ given the intersection array of $\Gamma$ is a direct problem, and finding the intersection array of $\Gamma$ given the parameters of $\Gamma_i$ is the inverse problem. The direct and inverse problems were solved earlier by A.A. Makhnev and M.S. Nirova for $i=3$. In the present paper, we solve the inverse problem for $i=2$: given the parameters of a strongly regular graph $\Gamma_2$, we find the intersection array of a distance-regular graph $\Gamma$ of diameter 3. It is proved that $\Gamma_2$ is not a graph in the half case. We also refine Nirova's results on distance-regular graphs $\Gamma$ of diameter 3 for which $\Gamma_2$ and $\Gamma_3$ are strongly regular. New infinite series of admissible intersection arrays are found: $\{r^2+3r+1,r(r+1),r+2;1,r+1,r(r+2)\}$ for odd $r$ divisible by 3 and $\{2r^2+5r+2,r(2r+2),2r+3;1,2r+2,r(2r+3)\}$ for $r$ indivisible by $3$ and not congruent to $\pm 1$ modulo $5$.
Keywords: strongly regular graph, distance-regular graph, intersection array.
Funding agency Grant number
Russian Science Foundation 14-11-00061-П
This work was supported by the Russian Science Foundation (project no. 14-11-00061-П).
Received: 11.05.2018
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2019, Volume 307, Issue 1, Pages S88–S98
DOI: https://doi.org/10.1134/S0081543819070071
Bibliographic databases:
Document Type: Article
UDC: 519.17
MSC: 05C25
Language: Russian
Citation: A. A. Makhnev, D. V. Paduchikh, “Inverse problems in distance-regular graphs theory”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 3, 2018, 133–144; Proc. Steklov Inst. Math. (Suppl.), 307, suppl. 1 (2019), S88–S98
Citation in format AMSBIB
\Bibitem{MakPad18}
\by A.~A.~Makhnev, D.~V.~Paduchikh
\paper Inverse problems in distance-regular graphs theory
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2018
\vol 24
\issue 3
\pages 133--144
\mathnet{http://mi.mathnet.ru/timm1557}
\crossref{https://doi.org/10.21538/0134-4889-2018-24-3-133-144}
\elib{https://elibrary.ru/item.asp?id=35511282}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2019
\vol 307
\issue , suppl. 1
\pages S88--S98
\crossref{https://doi.org/10.1134/S0081543819070071}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000451634900013}
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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