Remoteness and distance, distance (signless) Laplacian eigenvalues of a graph

Let G be a connected graph of order n. The remoteness of G, denoted by ρ, is the maximum average distance from a vertex to all other vertices. Let \(\partial_{1}\geq\cdots\geq\partial_{n}\), \(\partial_{1}^{L}\geq\cdots\geq\partial_{n}^{L}\) and \(\partial_{1} ^{Q}\geq\cdots\geq\partial_{n}^{Q}\) be the distance, distance Laplacian and distance signless Laplacian eigenvalues of G, respectively. In this paper, we give lower bounds on \(\rho+\partial _{1}\), \(\rho-\partial_{n}\), \(\rho+\partial_{1}^{L}\), \(\partial_{1} ^{L}-\rho\), \(2\rho+\partial_{1}^{Q}\) and \(\partial_{1}^{Q}-2\rho\) and the corresponding extremal graphs are also characterized.