The (d,1)-total labelling number λrd(G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and edges of G such that no two adjacent vertices or two adjacent edges have the same labels and the difference between the labels of a vertex and its incident edges is at least d.

The problem of(d,1)-total labeling number for the product of path and cycle graphs is studied,and some results of(d,1)-total labeling number for the product of path and cycle graphs are obtained.

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.