  Let $G$ be a simple connected graph, and $D(G)$ be the distance matrix of
$G$. Suppose that $D_{\max}(G)$ and $\lambda_1(G)$ are the maximum row sum and
the spectral radius of $D(G)$, respectively. In this paper, we give a lower
bound for $D_{\max}(G)-\lambda_1(G)$, and characterize the extremal graphs
attaining the bound. As a corollary, we solve a conjecture posed by Liu, Shu
and Xue.
