Graphs with Projective Linear Stabilizers

We investigate the structure of the free amalgamated productP1*P1∩P2P2in whichP1andP2are isomorphic projective linear groups andP1∩P2is a one-point stabilizer in the natural action ofPion the points of a projective space of dimensionn ≥ 2. We apply the results to graphs admitting a vertex-transitive automorphism group such that the subgroup stabilizing a vertex is a projective linear group. In particular, graphs of girth 4 with projective linear stabilizers are characterized.