On automorphisms of a strongly regular graph with parameters $(117,36,15,9)$

A. K. Gutnova ; A. A. Makhnev

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 4, pp. 43-49

Résumé

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 $117$.