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

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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. 
@article{UMJ_2020_6_2_a5,
     author = {Konstantin S. Efimov and Alexander A. Makhnev},
     title = {Distance-regular graph with intersection array $\{27, 20, 7; 1, 4, 21\}$ does not exist},
     journal = {Ural mathematical journal},
     pages = {63--67},
     publisher = {mathdoc},
     volume = {6},
     number = {2},
     year = {2020},
     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/