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

TL;DR

This paper investigates conjectures relating the Randić index and graph radius, proving new bounds using the harmonic index that improve previous results for various classes of graphs.

Contribution

The paper introduces bounds on the Randić index in terms of the graph radius using the harmonic index, strengthening existing conjectures for graphs with cyclomatic number and trees.

Findings

Proves R(G) ≥ r(G) - 31/105(k-1) for graphs with cyclomatic number k≥1.

Shows R(T) > r(T) + 1/15 for trees, excluding even paths.

Improves bounds on Randić index related to graph radius.

Abstract

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.