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DOI: 10.23671/VNC.2018.4.23386

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

Gutnova, A. K. , Makhnev A. A.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 4.
Abstract:
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 $$117$$.
Keywords: strongly regular graph, symmetric graph, automorphism groups of a graph.
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