A graph G in [l

m]-Path --connected if for every two venices of G

there is a k-vertex path connecting them

where k=l

l + 1

…

m. G is named P (k)-graph if d (x) + d (y) ≥k holds for every pairs of venices x and y of G with distance 2.In this paper we study the path-connectivity of a kind of p(k) -graph and make an improvement of Faudree-Schelp Theorem on path-connected graphs.Theorem 1. Let G= (V

E) be an n-vertex p(n -1)-graph. If G is [n-1

n]-path-connected

then G is [8

n]-path-connected (n≥8).Theorem 2. Let G= (V

E) be a 3-connected p(n)-graph n venices. If the independent number a(G) KeyWords：