A Proof of a Conjecture on the Distance Spectral Radius and Maximum Transmission of Graphs

Abstract

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.