图中的邻近度、偏远度与最大度

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.

Proximity and remoteness in triangle-free and C_4-free graphs in terms of order and minimum degree

Let $G$ be a finite, connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $ρ(G)$ and the proximity $π(G)$ of $G$ are the maximum and the minimum of the average distances of the vertices of $G$. In this paper, we present a sharp upper bound on the remoteness of a triangle-free graph of given order and minimum degree, and a corresponding bound on the proximity, which is sharp apart from an additive constant. We also present upper bounds on the remoteness and proximity of $C_4$-free graphs of giv…

图中的邻近度与遥远度：综述

Proximity and Remoteness in Graphs: a survey

本文综述自2006年提出以来图论中邻近度 $\pi$ 与遥远度 $\rho$ 概念的相关研究成果。

Proximity and Remoteness in Directed and Undirected Graphs

Let $D$ be a strongly connected digraph. The average distance $\barσ(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $ρ(D)$ and proximity $π(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $π(D)$ and $ρ(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $π(T)=ρ(T)$ if and only if …

On the difference between proximity and other distance parameters in triangle-free graphs and $C_4$-free graphs

The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $π(G)$ and the remoteness $ρ(G)$ of $G$ are the minimum and the maximum of the average distances of the vertices of $G$. In this paper, we give upper bounds on the difference between the remoteness and proximity, the diameter and proximity, and the radius and proximity of a triangle-free graph with given order and minimum degree. We derive the latter two results by first proving lower bounds on the proximity in terms of order, minimum degree…

平面图中的邻近性及其他距离参数

On Proximity and other Distance Parameters in Planar Graphs

本文研究平面图中邻近性、遥远性、半径和直径等距离参数之间的差的上界，针对极大平面图及给定连通度的图显著改进了已知界，并给出紧性示例。