Reciprocal degree distance and graph properties

The reciprocal degree distance (RDD), defined for a connected graph G as vertex-degree- weighted sum of the reciprocal distances, that is, RDD(G)=∑ _u≠v [Formula presented], where d _G (u) is the degree of the vertex u in the graph G and d _G (u,v) denotes the distance between two vertices u and v in the graph G. The reciprocal degree distance is a weight version of the Harary index, just as the degree distance is a weight version of the Wiener index. Finding sufficient conditions for graphs possessing certain properties is an important and meaningful problem. In this paper, we give sufficient conditions for a graph to be k-connected or β-deficient in terms of the reciprocal degree distance.