Geodesic
Distance-regular graph with intersection array $\{27, 20, 7; 1, 4, 21\}$ does not exist
Ural mathematical journal, Tome 6 (2020) no. 2, pp. 63-67 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the class of distance-regular graphs of diameter $3$ there are $5$ intersection arrays of graphs with at most $28$ vertices and noninteger eigenvalue. These arrays are $\{18, 14, 5; 1, 2, 144\}$, $\{18, 15, 9; 1, 1, 10\}$, $\{21, 16, 10; 1, 2, 12\}$, $\{24, 21, 3; 1, 3, 18\}$, and $\{27, 20, 7; 1, 4, 21\}$. Automorphisms of graphs with intersection arrays $\{18, 15, 9; 1, 1, 10\}$ and $\{24, 21, 3; 1, 3, 18\}$ were found earlier by A. A. Makhnev and D. V. Paduchikh. In this paper, it is proved that a graph with the intersection array $\{27, 20, 7; 1, 4, 21\}$ does not exist.
Keywords: distance-regular graph, graph $\Gamma$, with strongly regular graph $\Gamma_3$
Mots-clés : automorphism. 
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     author = {Konstantin S. Efimov and Alexander A. Makhnev},
     title = {Distance-regular graph with intersection array $\{27, 20, 7; 1, 4, 21\}$ does not exist},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UMJ_2020_6_2_a5/}
}
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Konstantin S. Efimov; Alexander A. Makhnev. Distance-regular graph with intersection array $\{27, 20, 7; 1, 4, 21\}$ does not exist. Ural mathematical journal, Tome 6 (2020) no. 2, pp. 63-67. http://geodesic.mathdoc.fr/item/UMJ_2020_6_2_a5/

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