On the limiting distribution of the metric dimension for random forests

Distorted metrics on trees and phylogenetic forests

We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree $T$ on $n$ leaves with a path metric $d$, consider the pairwise distances $\{d(u,v)\}$ between leaves. It is well known that these determine the tree and the $d$ length of all edges. Here we consider distortions $\d$ of $d$ such that for all leaves $u$ and $v$ it holds that $|d(u,v) - \d(u,v)| < f/2$ if either $d(u,v) < M$ or $\d(u,v) < M$, where $d$ satisfies $f \leq d(e) \leq g$ for all edges $e$. Given such distortions we show how to reconstruct in polynomial time a forest $T_1,...,…

The Forest Metrics for Graph Vertices

We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs, it takes into account all paths between vertices. Formally, it is defined as d(i,j)=q_{ii}+q_{jj}-q_{ij}-q_{ji}, where q_{ij} is the (i,j)-entry of the {\em relative forest accessibility matrix} Q(ε)=(I+εL)^{-1}, L is the Laplacian matrix of the (weighted) (multi)graph, and εis a positive parameter. By the matrix-forest theorem, the (i,j)-entry of the relative forest accessibility matrix of a graph provides the specific number of spanning rooted forests such that i and j belong to …

Hitting Time Quasi-metric and Its Forest Representation

Let $\hat m_{ij}$ be the hitting (mean first passage) time from state $i$ to state $j$ in an $n$-state ergodic homogeneous Markov chain with transition matrix $T$. Let $Γ$ be the weighted digraph whose vertex set coincides with the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. It holds that $$ \hat m_{ij}= q_j^{-1}\cdot \begin{cases} f_{ij},&\text{if }\;\; i\ne j,\\ q, &\text{if }\;\; i=j, \end{cases} $$ where $f_{ij}$ is the total weight of 2-tree spanning converging forests in $Γ$ that have one tree containing $i$ and the other tre…

通过势论计算双森林数量与随机割大小

Counting two-forests and random cut size via potential theory

本文利用势论方法，通过顶点数、边数和生成树数给出生成双森林数量的下界，并由此得到随机双森林平均割大小的上界。

The Matrix-Forest Theorem and Measuring Relations in Small Social Groups

We propose a family of graph structural indices related to the Matrix-forest theorem. The properties of the basic index that expresses the mutual connectivity of two vertices are studied in detail. The derivative indices that measure "dissociation," "solitariness," and "provinciality" of vertices are also considered. A nonstandard metric on the set of vertices is introduced, which is determined by their connectivity. The application of these indices in sociometry is discussed.

Canonical measures on metric graphs and a Kazhdan's theorem

We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of "hyperbolic measures" on universal covers of metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. We will prove a generalized version of this theorem for metric graphs, allowing any infinite Galois cover to replace the universal cover. We will show all such limiting measures satisfy a version of Gauss-Bonnet formula which, using the theory of v…