Geodesic
Graphs with projective suborbits
Izvestiya. Mathematics , Tome 39 (1992) no. 1, pp. 869-893.

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A positive answer is given to the problem of the existence of an upper bound for orders of stabilizers of vertices of connected graphs in vertex-transitive groups of automorphisms, if these stabilizers are finite and, under actions on adjacent vertices, contain the group $\mathbf{PSL}_n(q)$ as normal subgroups in its natural doubly transitive representation, $\operatorname{char}\mathbf F_q>3$ ($n$ and $q$ fixed). 
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V. I. Trofimov. Graphs with projective suborbits. Izvestiya. Mathematics , Tome 39 (1992) no. 1, pp. 869-893. http://geodesic.mathdoc.fr/item/IM2_1992_39_1_a8/

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