An L(j,k)-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive integers which are at least j apart,and vertices at distance two receive integers which are at least k apart.

The real L(j,k)-labeling of a graph G is a generalization of the L(j,k)-labeling.

Given an L(j,k)-labeling f of G,define the L(j,k) edge span of f,βj,k(G,f)=max{|f(x)-f(y)|: {x,y}∈E(G)}.

The real L(j,k) edge span of a graph G is defined accordingly,and is denoted by βj,k(G).

In this thesis, we firstly summarize the main results and evolution on the L(2,1)-labeling in recent years, then we investigate some graphs:(1) We explore the L(2,1)-labeling of cacti, and prove thatλ(G)≤△(G)+2 when△(G)≥5.