On the $D_{\alpha }$ spectral radius of non-transmission regular graphs

Let $G$ be a connected graph with order $n$ and size $m$. Let $D(G)$ and $Tr(G)$ be the distance matrix and diagonal matrix with vertex transmissions of $G$, respectively. For any real $\alpha \in [0,1]$, the generalized distance matrix $D_{\alpha }(G)$ of $G$ is defined as $$D_{\alpha }(G)=\alpha Tr(G)+(1-\alpha )D(G)\mathrm{.}$$ The largest eigenvalue of $D_{\alpha }(G)$ is called the $D_{\alpha }$ spectral radius or generalized distance spectral radius of $G$, denoted by ${\mu }_{\alpha }(G)$. In this paper, we establish a lower bound on the difference between the maximum vertex transmission and the $D_{\alpha }$ spectral radius of non-transmission regular graphs, and we also characterize the extremal graphs attaining the bound.