Geodesic
Distance-regular graph with intersection array $\{143,108,27;1,12,117\}$ does not exist
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 207-210

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There is a formally self-dual distance-regular graph $\Gamma$ with classical parameters $d=3$, $b=\alpha+1=q$, $\beta=q^2+q-1$ and intersection array $\{(q^2+q-1)(q^2+q+1),(q^2+q)q^2,q^3;1,(q^2+q),q^2(q^2+q+1)\}$. For the graph $\Gamma$ we have the strongly regular graphs $\Gamma_2$ and $\Gamma_3$ ($\Gamma_3$ is pseuqo-geometric for $pG_{q-1}(q^2+q-1,(q^2+q+1)(q-1))$). It is proved that a distance-regular graph with intersection array $\{143,108,27;1,12,117\}$ ($q=3$) does not exist.
Keywords: distance-regular graph, formally self-dual graph, triple intersection numbers. 
@article{SEMR_2023_20_1_a6,
     author = {A. A. Makhnev and M. M. Isakova and A. A. Tokbaeva},
     title = {Distance-regular graph with intersection array $\{143,108,27;1,12,117\}$ does not exist},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {207--210},
     publisher = {mathdoc},
     volume = {20},
     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2023_20_1_a6/}
}
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A. A. Makhnev; M. M. Isakova; A. A. Tokbaeva. Distance-regular graph with intersection array $\{143,108,27;1,12,117\}$ does not exist. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 207-210. http://geodesic.mathdoc.fr/item/SEMR_2023_20_1_a6/