On a conjecture of the Randić index and the minimum degree of graphs

The Randić index R(G)R(G) of a graph GG is defined by R(G)=∑uv1d(u)d(v), where d(u)d(u) is the degree of a vertex uu and the summation extends over all edges uvuv of GG. Delorme et al. (2002)  [6] put forward a conjecture concerning the minimum Randić index among allnn-vertex connected graphs with the minimum degree at least kk. In this work, we show that the conjecture is true given the graph contains kk vertices of degree n−1n−1. Further, it is true among kk-trees.