相关论文: Low-degree minimal spanning trees in normed spaces

Heydari \cite{heydari2013} presented very nice formulae for the Wiener and terminal Wiener indices of generalized Bethe trees. It is pity that there are some errors for the formulae. In this paper, we correct these errors and characterize…

We show that for every $n$-point metric space $M$ there exists a spanning tree $T$ with unweighted diameter $O(\log n)$ and weight $\omega(T) = O(\log n) \cdot \omega(MST(M))$. Moreover, there is a designated point $rt$ such that for every…

Minimal spanning trees (MSTs) have been used in cosmology and astronomy to distinguish distributions of points in a multi-dimensional space. They are essentially unknown in particle physics, however. We briefly define MSTs and illustrate…

The finite spectrum of a first-order sentence is the set of positive integers that are the sizes of its models. The class of finite spectra is known to be the same as the complexity class NE. We consider the spectra obtained by limiting…

This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the…

We study the problem of maximizing the number of full degree vertices in a spanning tree $T$ of a graph $G$; that is, the number of vertices whose degree in $T$ equals its degree in $G$. In cubic graphs, this problem is equivalent to…

The Hadamard rank of a point with respect to a projective variety is, if it exists, the minimum number of points of the variety whose coordinate-wise product is the given point. We classify the projective varieties for which the Hadamard…

This paper addresses the following questions for a given tree $T$ and integer $d\geq2$: (1) What is the minimum number of degree-$d$ subtrees that partition $E(T)$? (2) What is the minimum number of degree-$d$ subtrees that cover $E(T)$? We…

The Multimarginal Schr\"odinger Bridge (MSB) finds the optimal coupling among a collection of random vectors with known statistics and a known correlation structure. In the MSB formulation, this correlation structure is specified \emph{a…

Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f\) and let~\(K_n\) be the complete graph formed by joining each pair of nodes by a straight line…

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are…

In this note we prove two results on the quantitative illumination parameter f(d) of the unit ball of a d-dimensional normed space introduced by K. Bezdek (1992). The first is that f(d) = O(2^d d^2 log d). The second involves Steiner…

The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary…

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or…

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error e from 120 degrees.This notion arises in numerical approximations of…

We consider minimum-cost spanning trees, both in lattice and Euclidean models, in d dimensions. For the cost of the optimum tree in a box of size L, we show that there is a correction of order L^theta, where theta < 0 is a universal…

The necessary and sufficient conditions under which a given family $\mathcal{F}$ of subsets of finite set $X$ coincides with the family $\mathbf{B}_X$ of all balls generated by some ultrametric $d$ on $X$ are found. It is shown that the…

Consider the minimum spanning tree (MST) of the complete graph with n vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by n^{1/3} and with the uniform measure on its…

\emph{A maximal distance minimizer} for a given compact set $M \subset \mathbb{R}^2$ and some given $r > 0$ is a set having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset…

Let $T$ be a tree, a vertex of degree one is a \emph{leaf} of $T$ and a vertex of degree at least three is a \emph{branch vertex} of $T$. The {\it reducible stem } of $T$ is the smallest subtree that contains all branch vertices of $T$. In…