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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, Volume 28, Number 2, Pages 176–186
DOI: https://doi.org/10.21538/0134-4889-2022-28-2-176-186
(Mi timm1913)
 

On $Q$-polynomial Shilla graphs with $b = 4$

A. A. Makhnev, I. N. Belousov, M. P. Golubyatnikov

N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
References:
Abstract: Shilla graphs introduced by J. H. Koolen and J. Park are considered. In the problem of finding feasible intersection arrays of Shilla graphs with a fixed parameter $b$, $Q$-polynomial graphs play an important role. For such graphs, the smallest eigenvalue is the minimum possible for the third nonprincipal eigenvalue. Intersection arrays of $Q$-polynomial graphs were found for $b=3$ in 2010 by Koolen and Park and for $b\in\{4,5\}$ in 2018 by Belousov. In particular, it is known that a $Q$-polynomial Shilla graph with $b=4$ has intersection array $\{104,81,27;1,9,78\}$, $\{156,120,36;1,12,117\}$, or $\{20(q-2),3(5q-9),2q;1,2q,15(q-2)\}$, where $q=6,9,18$. We prove that distance-regular graphs with intersection arrays $\{80,63,12;1,12,60\}$, $\{140,108,18;1,18,105\}$, and $\{320,243,36;1,36,240\}$ do not exist.
Keywords: Shilla graph, distance-regular graphs, $Q$-polynomial graph.
Funding agency Grant number
Russian Science Foundation 19-71-10067
This work was supported by the Russian Science Foundation (project 19-71-10067).
Received: 15.03.2022
Revised: 15.04.2022
Accepted: 18.04.2022
Bibliographic databases:
Document Type: Article
UDC: 519.17
MSC: 05E30, 05C50
Language: Russian
Citation: A. A. Makhnev, I. N. Belousov, M. P. Golubyatnikov, “On $Q$-polynomial Shilla graphs with $b = 4$”, Trudy Inst. Mat. i Mekh. UrO RAN, 28, no. 2, 2022, 176–186
Citation in format AMSBIB
\Bibitem{MakBelGol22}
\by A.~A.~Makhnev, I.~N.~Belousov, M.~P.~Golubyatnikov
\paper On $Q$-polynomial Shilla graphs with $b = 4$
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2022
\vol 28
\issue 2
\pages 176--186
\mathnet{http://mi.mathnet.ru/timm1913}
\crossref{https://doi.org/10.21538/0134-4889-2022-28-2-176-186}
\elib{https://elibrary.ru/item.asp?id=48585958}
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