A proof of a conjecture on the distance spectral radius and maximum transmission of graphs

TL;DR

This paper establishes a lower bound for the difference between the maximum row sum and the spectral radius of a graph's distance matrix, characterizing extremal graphs and confirming a conjecture by Liu, Shu, and Xue.

Contribution

It provides a new lower bound for the difference between maximum transmission and spectral radius of the distance matrix, solving an open conjecture.

Findings

Lower bound for $D_{max}(G)-mbda_1(G)$ established

Extremal graphs attaining the bound characterized

Conjecture by Liu, Shu, and Xue confirmed

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.