On Proximity Measures for Graph Vertices

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 …

Uniqueness of best proximity pairs and rigidity of semimetric spaces

For arbitrary semimetric space $(X, d)$ and disjoint proximinal subsets $A$, $B$ of $X$ we define the proximinal graph as a bipartite graph with parts $A$ and $B$ whose edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A, B)$. We characterize the semimetric spaces whose proximinal graphs have at most one edge and the semimetric spaces whose proximinal graphs have the vertices with degree at most $1$ only. This allows us to describe the necessary and sufficient conditions for uniqueness of the best proximity pairs and best approximations.

有界面外平面图中的邻近度与半径

Proximity and Radius in Outerplanar Graphs with Bounded Faces

本文研究了有界面外平面图的邻近度和半径，给出了2-连通外平面图邻近度的紧上界，并将半径上界推广到最大面长度受限的更大图类。

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。

Similarities on Graphs: Kernels versus Proximity Measures

We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This helps to understand the mathematical nature of such measures and can potentially be useful for recommending the adoption of specific similarity measures in data analysis.

Bipartite graphs and best proximity pairs

We say that a bipartite graph $G(A, B)$ with fixed parts $A$, $B$ is proximinal if there is a semimetric space $(X, d)$ such that $A$ and $B$ are disjoint proximinal subsets of $X$ and all edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A, B)$. It is proved that a bipartite graph $G$ is not isomorphic to any proximinal graph iff $G$ is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph $G$ is a disjoint union of complete bipartite graphs iff $G$ is isomorphic to a nonempty proximinal graph for an ultramet…

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。

Proximity Induced by Order Relations

This paper introduces an order proximity on a collection of objects induced by a partial order using the Smirnov closeness measure on a Száz relator space. A Száz relator is a nonempty family of relations defined on a nonvoid set $K$. The Smirnov closeness measure provides a straightforward means of assembling partial ordered of pairwise close sets. In its original form, Ju. M. Smirnov closeness measure $δ(A,B) = 0$ for a pair of nonempty sets $A,B$ with nonvoid intersection and $δ(A,B) = 1$ for non-close sets. A main result in this paper is that the graph obtained by the proximity is equivale…

How tree-based is my network? Proximity measures for unrooted phylogenetic networks

Tree-based networks are a class of phylogenetic networks that attempt to formally capture what is meant by "tree-like" evolution. A given non-tree-based phylogenetic network, however, might appear to be very close to being tree-based, or very far. In this paper, we formalise the notion of proximity to tree-based for unrooted phylogenetic networks, with a range of proximity measures. These measures also provide characterisations of tree-based networks. One measure in particular, related to the nearest neighbour interchange operation, allows us to define the notion of "tree-based rank". This pro…

系统发生网络类别的近似度量

Proximity Measures for Classes of Phylogenetic Networks

研究系统发生网络类别，考虑基于叶添加、有效弧删除和弧删除的三种近似度量，研究其成对可比性、复杂性结果及网络类别的极值界限，以衡量给定网络与特定类别接近程度。

顶点位于无限平面测度集合中的单位面积等腰梯形

Isosceles trapezoids of unit area with vertices in sets of infinite planar measure

本文肯定回答了 Erdős 的问题：每个无限 Lebesgue 测度的可测平面集必含单位面积等腰梯形的四个顶点，并将肯定结论推广到等腰三角形和直角三角形。

Spanning Forests and the Golden Ratio

For a graph G, let f_{ij} be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f_{ij}'s can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The {\em doubly stochastic graph matrix} is the matrix F=(f_{ij})/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F^{-1}=I+L, where L is the Laplacian matrix of G. We show that fo…

面积为1的多边形及其顶点位于平面无限测度集合中

Polygons of unit area with vertices in sets of infinite planar measure

研究探讨了面积为1的多边形顶点在无限测度集合中的存在性，分别回答了循环四边形和等边凸多边形的问题。

New Characterisations of Tree-Based Networks and Proximity Measures

Phylogenetic networks are a type of directed acyclic graph that represent how a set $X$ of present-day species are descended from a common ancestor by processes of speciation and reticulate evolution. In the absence of reticulate evolution, such networks are simply phylogenetic (evolutionary) trees. Moreover, phylogenetic networks that are not trees can sometimes be represented as phylogenetic trees with additional directed edges placed between their edges. Such networks are called {\em tree based}, and the class of phylogenetic networks that are tree based has recently been characterised. In …