The Randic index and the diameter of graphs

TL;DR

This paper proves a conjecture that among all connected graphs with n vertices, the path graph minimizes the ratio and difference of the Randić index and diameter, establishing a new extremal property.

Contribution

The paper proves the conjecture that the path graph uniquely minimizes the Randić index to diameter ratio and difference among connected graphs, and establishes a stronger inequality involving these parameters.

Findings

Path graphs uniquely minimize R(G)/D(G) and R(G)-D(G).

Established a lower bound for R(G)-(1/2)D(G).

Proved the conjecture completely.

Abstract

The {\it Randi\'c index} $R(G)$ of a graph $G$ is defined as the sum of 1/\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v,$ respectively. Let $D(G)$ be the diameter of $G$ when $G$ is connected. Aouchiche-Hansen-Zheng conjectured that among all connected graphs $G$ on $n$ vertices the path $P_n$ achieves the minimum values for both $R(G)/D(G)$ and $R(G)- D(G)$. We prove this conjecture completely. In fact, we prove a stronger theorem: If $G$ is a connected graph, then $R(G)-(1/2)D(G)\geq \sqrt{2}-1$, with equality if and only if $G$ is a path with at least three vertices.