On Automorphisms of a Strongly Regular Graph with Parameters \((117,36,15,9)\)

In the previous work of the authors  some arrays of intersections of distance-regular graphs were found, in which the neighborhoods of the vertices are pseudogeometric graphs for  \(pG_{s-3}(s,t)\). In particular, a locally pseudo \(pG_2(5,2)\)-graph is a strongly regular  graph with parameters \((117,36,15,9)\). The main result of this paper gives a description of possible orders and the structure of the subgraphs of fixed points of automorphisms of a strongly regular graph with parameters \((117,36,15,9)\). 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 this  result, the following corollary 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 \(117\).

For citation: Gutnova, A. K. and Makhnev, A. A. On Automorphisms of a Strongly Regular Graph with Parameters \((117,36,15,9)\), Vladikavkaz Math. J., 2018, vol. 20, no. 4, pp. 43-49 (in Russian). DOI 10.23671/VNC.2018.4.23386

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