Abstract: Let G=(V,E) be a graph. If φ is a function from the vertex set V(G) to the set of positive integers. Then two vertices u, v ∈ V(G) are φ -equitable if｜φ(u)－φ(v)｜≤1.By the degree, equitable adjacency between vertices can be redefine almost all of the variants of the graphs. In this paper we study the degree equitability of the graph by defining equitable connectivity, equitable regularity, equitable connected graph and equitable complete graph. Some new families of graphs and some interesting results are obtained.