Abstrakt
The Randic index R(G) of a nontrivial connected graph G is defined as the sum of the weights (d(u)d(v))^(-0.5) over all edges e=uv of G. We prove that R(G) }= d(G)/2, where d(G) is the diameter of G.
This immediately implies that R(G) }= r(G)/2, which is the closest result to the well-known Graffiti conjecture R(G)}= r(G) - 1 of Fajtlowicz, where r(G) is the radius of G. Asymptotically, our result approaches the bound R(G)/d(G) }= (n-3+2*sqrt(2))/(2n-2) conjectured by Aouchiche, Hansen and Zheng.