On a Conjecture of Randi\'{c} Index and Graph Radius

The Randi\'{c} index $R(G)$ of a graph $G$ is defined as the sum of $(d_i d_j)^{-1/2}$ over all edges $v_i v_j$ of $G$, where $d_i$ is the degree of the vertex $v_i$ in $G$. The radius $r(G)$ of a graph $G$ is the minimum graph eccentricity of any graph vertex in $G$. Fajtlowicz(1988) conjectures $R(G) \ge r(G)-1$ for all connected graph $G$. A stronger version, $R(G) \ge r(G)$, is conjectured by Caporossi and Hansen(2000) for all connected graphs except even paths. In this paper, we make use of Harmonic index $H(G)$, which is defined as the sum of $\frac{2}{d_i+d_j}$ over all edges $v_i v_j$ of $G$, to show that $R(G) \ge r(G)-31/105(k-1)$ for any graph with cyclomatic number $k\ge 1$, and $R(T)> r(T)+1/15$ for any tree except even paths. These results improve and strengthen the known results on these conjectures.