Geodesic
Distance-regular Terwilliger graphs with intersection arrays $\{50,42,1;1,2,50\}$ and $\{50,42,9;1,2,42\}$ do not exist
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1075-1082.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\Gamma$ be a distance-regular graph and its local subgraphs are isomorphic the Hoffman-Singleton graph. A.L. Gavrilyuk and A.A. Makhnev proved that $\Gamma$ is the Terwilliger graph with intersection array $\{50,42,9;1,2,42\}$ or $\{50,42,1;1,2,50\}$. In this paper we prove that Terwilliger graphs with intersection arrays $\{50,42,1;1,2,50\}$ and $\{50,42,9;1,2,42\}$ do not exist.
Keywords: distance-regular graph, Terwilliger graph, triple intersection numbers. 
@article{SEMR_2021_18_2_a39,
     author = {A. A. Makhnev and M. S. Nirova},
     title = {Distance-regular {Terwilliger} graphs with intersection arrays $\{50,42,1;1,2,50\}$ and $\{50,42,9;1,2,42\}$ do not exist},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1075--1082},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a39/}
}
TY  - JOUR
AU  - A. A. Makhnev
AU  - M. S. Nirova
TI  - Distance-regular Terwilliger graphs with intersection arrays $\{50,42,1;1,2,50\}$ and $\{50,42,9;1,2,42\}$ do not exist
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2021
SP  - 1075
EP  - 1082
VL  - 18
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a39/
LA  - ru
ID  - SEMR_2021_18_2_a39
ER  - 
%0 Journal Article
%A A. A. Makhnev
%A M. S. Nirova
%T Distance-regular Terwilliger graphs with intersection arrays $\{50,42,1;1,2,50\}$ and $\{50,42,9;1,2,42\}$ do not exist
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2021
%P 1075-1082
%V 18
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a39/
%G ru
%F SEMR_2021_18_2_a39
A. A. Makhnev; M. S. Nirova. Distance-regular Terwilliger graphs with intersection arrays $\{50,42,1;1,2,50\}$ and $\{50,42,9;1,2,42\}$ do not exist. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1075-1082. http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a39/

[1] A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-regular graphs, Springer-Verlag, Berlin etc, 1989 | Zbl

[2] A.A. Makhnev, “Graphs in which neighborhoods of vertices are isomorphic to the Hoffman-Singleton graph”, Proc. Steklov Inst. Math., 267, Suppl. 1 (2009), S128–S148 | Zbl

[3] A.L. Gavrilyuk, A.A. Makhnev, “On distance-regular graphs in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph”, Dokl. Math., 80:2 (2009), 665–668 | DOI | Zbl

[4] A.L. Gavrilyuk, Wenbin Guo, A.A. Makhnev, “Automorphisms of Terwilliger graphs with $\mu = 2$”, Algebra Logik, 47:5 (2008), 330–339 | DOI | Zbl

[5] K. Coolsaet, A. Jurišić, “Using equality in the Krein conditions to prove nonexistence of sertain distance-regular graphs”, J. Comb. Theory, Ser. A, 115:6 (2008), 1086–1095 | DOI | Zbl