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

Abstract

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 $$D_{\alpha }(G)=\alpha Tr(G)+(1-\alpha )D(G).$$ 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.

This article has undergone further revisions and enhancements, all of which were contributed by Ligong Wang. In recognition of his contributions, Ligong Wang is now acknowledged as a new co-author. The manuscript has been updated to its current version to incorporate these changes. We affirm that all authors have reviewed and approved this update