About a Conjecture on the Randic Index of Graphs

For an edge uv of a graph G, the weight of the edge e = uv is defined by w(e) = 1/ √ d(u)d(v). Then R(G) = ∑_uvϵE(G) 1/ √ d(u)d(v)=∑_uvϵE(G) w(e) is called the Randic index of G. If G is a connected graph, then rad(G) = min_x max_y d(x;y) is called the radius of G, where d(x,y) is the distance between two vertices x;y. In 2000, Caporossi and Hansen conjectured that for all connected graphs except the even paths, R(G) ≥ r(G). They proved the conjecture holds for all trees except the even paths. In this paper, it is proved that the conjecture holds for all unicyclic graphs, bicyclic graphs and some class of chemical graphs.