1.vertex v of graph G= (V, E) is called weak-localy connected, if there isa vertex u∈V-{v} such that the distance between any pair of venices in induced subgraphG[N(v) U {u} ] is at most 3. Let G be a nontrivial connected graph with no induced claw.
4.This paper indicates, if the independence number of vertices at a distance 2 from a vertex in G is one, where G is 2—connected graph, then G is Hamilton graph.
5.Like the above result, any two vertices which are of distance 2 is discussed, then we obtain the following result in this thesis: let G be a graph of order n ≥ 3 such that d(u) + d(v) ≥ [(4n)/3] - 1 where d(u, v) = 2 in G, then every vertex of G is contained in a 3-cycle.
