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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.

In this research work, we introduce a new version of leap Zagreb index called fourth leap Zagreb index. Mathematical properties and exact values for some standard classes of graphs for this new topological index are obtained. Also, the upper bounds are established. The values of this index for some graph operations containing the Cartesian product, Join and Composition are found.

The second degree of a vertex in a simple graph is defined as the number of its second neighbors. The leap eccentric connectivity index of a graph M , L ξ c M , is the sum of the product of the second degree and the eccentricity of every vertex in M . In this paper, some lower and upper bounds of L ξ c S M in terms of the numbers of vertices and edges, diameter, and the first Zagreb and third leap Zagreb indices are obtained. Also, the exact values of L ξ c S M for some well-known graphs are computed.

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