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Sib. Èlektron. Mat. Izv., 2019, Volume 16, Pages 206–216 (Mi semr1050)  

Discrete mathematics and mathematical cybernetics

Distance-regular graph with intersection array $\{105,72,24;1,12,70\}$ does not exist

I. N. Belousov, A. A. Makhnev

N.N. Krasovskii Institute of Mathematics and Mechanics of UB RAS, 16, S.Kovalevskaya str., Yekaterinburg, 620990, Russia

Abstract: Distance-regular graph $\Gamma$ of diameter 3 is called Shilla graph if $\Gamma$ containes the second eigenvalue $\theta_1=a_3$. In this case $a=a_3$ devides $k$ and we set $b=b(\Gamma)=k/a$. Koolen and Park obtained the list of intersection arrays for Shilla graphs with $b=3$. A. Brouwer with coauthors proved that graph with intersection array $\{27,20,10;1,2,18\}$ does not exist. $Q$-polinomial Shilla graph with $b=3$ has intersection array $\{42,30,12;1,6,28\}$ or $\{105,72,24;1,12,70\}$. Early authors proved that graph with intersection array $\{42,30,12;1,6,28\}$ does not exist.
We prove that graph with intersection array $\{105,72,24;1,12,70\}$ does not exist.

Keywords: distance-regular graph, Shilla graph, triple intersection numbers.

DOI: https://doi.org/10.33048/semi.2019.16.012

Full text: PDF file (173 kB)
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Bibliographic databases:

Document Type: Article
UDC: 519.17
MSC: 05C25
Received December 18, 2018, published February 8, 2019

Citation: I. N. Belousov, A. A. Makhnev, “Distance-regular graph with intersection array $\{105,72,24;1,12,70\}$ does not exist”, Sib. Èlektron. Mat. Izv., 16 (2019), 206–216

Citation in format AMSBIB
\Bibitem{BelMak19}
\by I.~N.~Belousov, A.~A.~Makhnev
\paper Distance-regular graph with intersection array $\{105,72,24;1,12,70\}$ does not exist
\jour Sib. \`Elektron. Mat. Izv.
\yr 2019
\vol 16
\pages 206--216
\mathnet{http://mi.mathnet.ru/semr1050}
\crossref{https://doi.org/10.33048/semi.2019.16.012}


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