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 ∂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.
Remoteness, proximity and few other distance invariants in graphs
We establish maximal trees and graphs for the difference of average distance and proximity proving thus the corresponding conjecture posed in [4]. We also establish maximal trees for the difference of average eccentricity and remoteness and minimal trees for the difference of remoteness and radius proving thus that the corresponding conjectures posed in [4] hold for trees.
Distance spectral radius of trees with fixed maximum degree
C ‡ Abstract. Distance energy is a newly introduced molecular graph-based analog of the total �-electron energy, and it is defined as the sum of the absolute eigenvalues of the molecular distance matrix. For trees and unicyclic graphs, distance energy is equal to the doubled value of the distance spectral radius. In this paper, we introduce a general transformation that increases the distance spectral radius and provide an alternative proof that the path Pn has the maximal distance spectral radius among trees on n vertices. Among the trees with a fixed maximum degree �, we prove that the broom Bn,� (consisting of a star S�+1 and a path of length n � 1 attached to an arbitrary pendent vertex of the star) is the unique tree that maximizes the distance spectral radius, and conjecture the structure of a tree which minimizes the distance spectral radius. As a first step towards this conjecture, we characterize the starlike trees with the minimum distance spectral radius.