Abstract
In this paper, we consider labelings of graphs in which the label on an edge is the absolute value of the difference of its vertex labels. Such a labeling using {0,1,2, ... ,k - 1} is called k-equitable if the number of vertices ( resp. edges) labeled i and the number of vertices (resp. edges) labeled j differ by at most one and is called k-balanced if the number of vertices labeled i and the number of edges labeled j differ by at most one. We determine which graphs in certain families are k-equitable or k-balanced and we give also some necessary conditions on these two labelings.