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

Abstract

Abstract 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 ∂1≥⋯≥∂n $\partial_{1}\geq\cdots\geq\partial_{n}$, ∂1L≥⋯≥∂nL $\partial_{1}^{L}\geq\cdots\geq\partial_{n}^{L}$ and ∂1Q≥⋯≥∂nQ $\partial_{1} ^{Q}\geq\cdots\geq\partial_{n}^{Q}$ be the distance, distance Laplacian and distance signless Laplacian eigenvalues of G, respectively. In this paper, we give lower bounds on ρ+∂1 $\rho+\partial _{1}$, ρ−∂n $\rho-\partial_{n}$, ρ+∂1L $\rho+\partial_{1}^{L}$, ∂1L−ρ $\partial_{1} ^{L}-\rho$, 2ρ+∂1Q $2\rho+\partial_{1}^{Q}$ and ∂1Q−2ρ $\partial_{1}^{Q}-2\rho$ and the corresponding extremal graphs are also characterized.

Keywords

• Remoteness
• Distance eigenvalues
• Distance (signless) Laplacian eigenvalues