  Let $G$ be a simple connected graph of order $n$ and $\partial(G)$ is the
spectral radius of the distance matrix $D(G)$ of $G$. The transmission $D_i$ of
vertex $i$ is the $i$-th row sum of $D(G)$. Denote by $D_{\max}(G)$ the maximum
of transmissions over all vertices of $G$, and $\partial^Q(G)$ is the spectral
radius of the distance signless Laplacian matrix
$D(G)+\mbox{diag}(D_1,D_2,\ldots,D_n)$. In this paper, we present a sharp lower
bound of $2D_{\max}(G)-\partial^Q(G)$ among all $n$-vertex connected graphs,
and characterize the extremal graphs. Furthermore, we give the minimum values
of respective $D_{\max}(G)-\partial(G)$ and $2D_{\max}(G)-\partial^Q(G)$ on
trees and characterize the extremal trees.
