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.
Categories: math.CO