The variation of the Randić index with regard to minimum and maximum degree

The variation of the Randić index R′(G) of a graph G is defined by  R′(G)=∑uv∈E(G)1max{d(u),d(v)}, where d(u) is the degree of vertex u and the summation extends over all edges uv of G. Let G(k,n) be the set of connected simple n-vertex graphs with minimum vertex degree k. In this paper we found in G(k,n) graphs for which the variation of the Randić index attains its minimum value. When k≤n2 the extremal graphs are complete split graphs Kk,n−k∗, which have only vertices of two degrees, i.e. degree k and degree n−1, and the number of vertices of degree k is n−k, while the number of vertices of degree n−1 is k. For k≥n2 the extremal graphs have also vertices of two degrees k and n−1, and the number of vertices of degree k is n2. Further, we generalized results for graphs with given maximum degree.