The linear programming approach to the Randić index

Let G(k, n) be the set of simple graphs (i.e. without multiple edges or loops) that have n vertices and the minimum degree of vertices is k. The Randi? index of a graph G is: , where δ_u is the degree of vertex u and the summation extends over all edges (uv) of G. Using linear programming, we find the extremal graphs or give good bounds for this index when the number n_k of vertices of degree kis n?k+t, for 0tk and kn/2. We also prove that for n_kn?k, (kn/2) the minimum value of the Randi? index is attained for the graph .