Distance Laplacian Spectral Radius of the Complements of Trees and Unicyclic Graphs

Abstract

Let $G$ be a connected graph and $D^{L}(G) = \operatorname{Tr}(G) - D(G)$ be the distance Laplacian matrix of $G$, where $\operatorname{Tr}(G)$ and $D(G)$ are diagonal matrix with vertex transmissions of $G$ and distance matrix of $G$, respectively. The $D^{L}$-spectral radius of $G$ is defined as the largest absolute value of the distance Laplacian eigenvalues of $G$. In this paper, we characterize the unique extremal graphs which maximize the $D^{L}$-spectral radius among the complements of trees and unicyclic graphs, respectively.