Geodesic
The nonexistence small $Q$-polynomial graphs of type (III)
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1270-1279

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I.N. Belousov, A.A. Makhnev and M.S. Nirova found the description of $Q$-polynomial distance-regular graphs $\Gamma$ of diameter 3 such that $\Gamma_2$ and $\Gamma_3$ are strongly regular. Such graph has intersection array $\{t(c_2+1)+a_3,tc_2,a_3+1;1,c_2,t(c_2+1)\}$ and $(c_2+1)=a_3(a_3+1)/(t^2-a_3-1)$. $Q$-polynomial graph $\Gamma$ is the graph of type (I), if $a_3$ is devided by $c_2+1$, graph of type (II), if $a_3+1$ is devided by $c_2+1$, graph of type (III), if $a_3$ and $a_3+1$ does not devided by $c_2+1$. In this paper it is proved that graph of type (III) with $t\le 6$ has intersection array $\{14,10,3;1,5,12\}$, $\{69,56,10;1,14,60\}$, $\{74,54,15;1, 9,60\}$, $\{87,66,16;1,11,72\}$, $\{119,100,15;1,20,105\}$ or $\{188,162,21;1, 27,168\}$. Further it is proved that graphs of type (III) with intersection array $\{14,10,3;1,5,12\}$, $\{87,66,16;1,11,72\}$ and $\{188,162,21;1,27,168\}$ do not exist.
Keywords: distance-regular graph, triple intersection numbers.
Mots-clés : $Q$-polynomial graph 
@article{SEMR_2020_17_a73,
     author = {A. A. Makhnev and M. M. Isakova and A. A. Tokbaeva},
     title = {The nonexistence small $Q$-polynomial graphs of type {(III)}},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1270--1279},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a73/}
}
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A. A. Makhnev; M. M. Isakova; A. A. Tokbaeva. The nonexistence small $Q$-polynomial graphs of type (III). Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1270-1279. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a73/