On a conjecture of the Randić index

The Randić index of a graph GG is defined as R(G)=∑u∼v(d(u)d(v))−12, where d(u)d(u) is the degree of vertex uu and the summation goes over all pairs of adjacent vertices uu, vv. A conjecture on R(G)R(G) for connected graph GG is as follows: R(G)≥r(G)−1R(G)≥r(G)−1, where r(G)r(G) denotes the radius of GG. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10n≤10 and tricyclic graphs.