A characterization of extremal non-transmission-regular graphs by the distance (signless Laplacian) spectral radius

Let $G$ be a simple connected graph of order $n$ and $\mathrm{\partial }(G)$ 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 ${\mathrm{\partial }}^Q(G)$ is the spectral radius of the distance signless Laplacian matrix $D(G)+diag(D_1,D_2,\dots ,D_n)$. In this paper, we present a sharp lower bound of $2D_{\max }(G)-{\mathrm{\partial }}^Q(G)$ among all $n$-vertex connected graphs, and characterize the extremal graphs. Furthermore, we give the minimum values of respective $D_{\max }(G)-\mathrm{\partial }(G)$ and $2D_{\max }(G)-{\mathrm{\partial }}^Q(G)$ on trees and characterize the extremal trees.