The Randi\'{c} index and signless Laplacian spectral radius of graphs

TL;DR

This paper proves a conjecture relating the Randić index and the signless Laplacian spectral radius of connected graphs, confirming the proposed bounds and characterizing the extremal graphs for all graph sizes.

Contribution

The paper completely resolves a conjecture on the ratio of the signless Laplacian spectral radius to the Randić index for all connected graphs.

Findings

Confirmed the conjectured bounds for all graph sizes.

Identified extremal graphs achieving equality.

Extended previous partial verifications to a complete proof.

Abstract

Given a connected graph $G$, the Randi\'c index $R(G)$ is the sum of $\tfrac{1}{\sqrt{d(u)d(v)}}$ over all edges $\{u,v\}$ of $G$, where $d(u)$ and $d(v)$ are the degree of vertices $u$ and $v$ respectively. Let $q(G)$ be the largest eigenvalue of the singless Laplacian matrix of $G$ and $n=|V(G)|$. Hansen and Lucas (2010) made the following conjecture: \[ \frac{q(G)}{R(G)} \leq \begin{cases} \frac{4n-4}{n} & 4 \leq n\leq 12 \frac{n}{\sqrt{n-1}} & n\geq 13 \end{cases} \] with equality if and only if $G=K_{n}$ for $4\leq n\leq 12$ and $G=S_n$ for $n\geq 13$, respectively. Deng, Balachandran, and Ayyaswamy (J. Math. Anal. Appl. 2014) verified this conjecture for $4 \leq n \leq 11$. In this paper, we solve this conjecture completely.