Geodesic
Distance-regular graph with intersection array $\{105,72,24;1,12,70\}$ does not exist
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 206-216

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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. 
@article{SEMR_2019_16_a62,
     author = {I. N. Belousov and A. A. Makhnev},
     title = {Distance-regular graph with intersection array $\{105,72,24;1,12,70\}$ does not exist},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {206--216},
     publisher = {mathdoc},
     volume = {16},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a62/}
}
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I. N. Belousov; A. A. Makhnev. Distance-regular graph with intersection array $\{105,72,24;1,12,70\}$ does not exist. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 206-216. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a62/