Dominating functions in graphs – regularity versus irregularity

A vertex v in a graph G is said to dominate a vertex u if either u = v or and a set S of vertices in G is a dominating set of G if every vertex of G is dominated by at least one vertex in S. Domination has been looked at in an equivalent way. A function is a dominating function of a graph G if for every vertex v of G. We use dominating functions to investigate graphs all of whose vertices are dominated by the same number of vertices as well as those graphs whose vertices are dominated by as many different number of vertices as possible.