Geodesic
On automorphisms of a strongly regular graph with parameters $(117,36,15,9)$
Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 4, pp. 43-49 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the works of A. A. Makhnev and A. K. Gutnova arrays of intersections of distance-regular graphs in which the neighborhoods of the vertices are pseudogeometric graphs for $pG_{s-3}(s,t)$ were found. In particular, the locally pseudo $pG_2(5,2)$-graph is a strongly regular graph with parameters $(117,36,15,9)$. The first main result of this paper is a theorem in which the possible orders and the structure of the subgraphs of fixed points of automorphisms of a strongly regular graph with parameters $(117,36,15,9)$ are found. This graph has a spectrum of $36^1,9^26,-3^90$. The order of clicks in $\Gamma$ does not exceed $1+36/3=13$, the order of the cocliques in $\Gamma$ does not exceed $117\cdot 3/39=9$. Further, from the obtained theorem, the following result is derived: if the group $\Gamma$ of automorphisms of a strongly regular graph with parameters $(117,36,15,9)$ acts transitively on the set of vertices, then the socle $T$ of the group $\Gamma$ is isomorphic to either $L_3(3)$ and $T_a\cong GL_2(3)$ is a subgroup of index $117$, or $T_a\cong GL_2(3)$ and $T_a\cong U_4(2).Z_2$ is a subgroup of index $117$. 
@article{VMJ_2018_20_4_a3,
     author = {A. K. Gutnova and A. A. Makhnev},
     title = {On automorphisms of a strongly regular graph with parameters $(117,36,15,9)$},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {43--49},
     year = {2018},
     volume = {20},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2018_20_4_a3/}
}
TY  - JOUR
AU  - A. K. Gutnova
AU  - A. A. Makhnev
TI  - On automorphisms of a strongly regular graph with parameters $(117,36,15,9)$
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2018
SP  - 43
EP  - 49
VL  - 20
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VMJ_2018_20_4_a3/
LA  - ru
ID  - VMJ_2018_20_4_a3
ER  - 
%0 Journal Article
%A A. K. Gutnova
%A A. A. Makhnev
%T On automorphisms of a strongly regular graph with parameters $(117,36,15,9)$
%J Vladikavkazskij matematičeskij žurnal
%D 2018
%P 43-49
%V 20
%N 4
%U http://geodesic.mathdoc.fr/item/VMJ_2018_20_4_a3/
%G ru
%F VMJ_2018_20_4_a3
A. K. Gutnova; A. A. Makhnev. On automorphisms of a strongly regular graph with parameters $(117,36,15,9)$. Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 4, pp. 43-49. http://geodesic.mathdoc.fr/item/VMJ_2018_20_4_a3/

[1] Gutnova A. K., Makhnev A. A., “Completely Regular Graphs in Which Neighborhoods of Vertices are Pseudogeometric Graphs for $pG_{s-3}(s, t)$”, Doklady Mathematics, 454:2 (2014), 145–148 (in Russian) | DOI | MR | Zbl

[2] Gutnova A. K., Makhnev A. A., “Locally Pseudo $GQ (4, t)$-Graphs”, Doklady Mathematics, 462:6 (2015), 637–641 (in Russian) | DOI | MR | Zbl

[3] Gutnova A. K., Makhnev A. A., “Graphs of Diameter not Greater than 3, in Which Neighborhoods of Vertices are Pseudogeometric Graphs for $pG_{s-3}(s, t)$”, Doklady Mathematics, 461:6 (2015), 629–632 (in Russian) | DOI | MR | Zbl

[4] Cameron P. J., Permutation Groups, London Math. Soc. Student Texts, 45, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[5] Gavrilyuk A. L., Makhnev A. A., “On Automorphisms of Amply Regular Graphs with the Intersection Array $\{56,45,1;1,9,56\}$”, Doklady Mathematics, 432:5, 512–515 (in Russian) | MR

[6] Zavarnitsine A. V., “Finite simple groups with narrow prime spectrum”, Sibirean Electr. Math. Reports, 6 (2009), 1–12 | MR | Zbl