In this paper, we consider the action of G= PSL(n,p) (n \geq 3, p is an odd prime, and (n, p-1)= 1) on the set of frames of the projective space PG(n - 1, p) over thefield GF(P), and survey the non-self-paired orbitals of G. Then we find an infinite family of vertex-primitive 1/2-transitive graphs. In addition, we get a method of computing the suborbits of PSL(n,p) acting on the frame-set of PG(n-1,p).