Related papers: Squared distance matrix of a weighted tree
Let T be a (not necessarily positive) weighted tree with n leaves numbered by the set {1,...,n}. Define the k-weights of the tree D_{i_1,....,i_k}(T) as the sum of the lengths of the edges of the minimal subtree connecting i_1,....,i_k. We…
We study the concept of the continuous mean distance of a weighted graph. For connected unweighted graphs, the mean distance can be defined as the arithmetic mean of the distances between all pairs of vertices. This parameter provides a…
Given two binary trees on $N$ labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartets distance between the two trees is…
The distance matrix of a connected graph is defined as the matrix in which the entries are the pairwise distances between vertices. The distance spectrum of a graph is the set of eigenvalues of its distance matrix. A graph is said to be…
Given a rooted tree $T$ with leaves $v_1,v_2,\ldots,v_n$, we define the ancestral matrix $C(T)$ of $T$ to be the $n \times n$ matrix for which the entry in the $i$-th row, $j$-th column is the level (distance from the root) of the first…
Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming…
We derive two formulas for the weighted sums of rooted spanning forests of particular sequence of graphs by using the matrix tree theorem. We consider cycle graphs with edges so called the pendant edges. One of our formula can be described…
A metric phylogenetic tree relating a collection of taxa induces weighted rooted triples and weighted quartets for all subsets of three and four taxa, respectively. New intertaxon distances are defined that can be calculated from these…
The pseudodeterminant $\textrm{pdet}(M)$ of a square matrix is the last nonzero coefficient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant. If $\partial$ is a symmetric or skew-symmetric matrix then…
Given a graph $G$, the exponential distance matrix is defined entry-wise by letting the $(u,v)$-entry be $q^{\text{dist}(u,v)}$, where $\text{dist}(u,v)$ is the distance between the vertices $u$ and $v$ with the convention that if vertices…
We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x…
We obtain an exact formula for the probability that a uniformly random spanning tree of the $2$-by-$n$ square grid is ``balanced'' in the sense that it has an edge whose removal partitions its vertices into two sets of equal size. We…
In this article we study the structured distance to singularity for a nonsingular matrix $A\in\mathbb{C}^{n\times n}$, with a prescribed linear structure $\mathcal{S}$ (for instance, a sparsity pattern, or a real Toeplitz structure), i.e.,…
We report on some findings concerning Connes' noncommutative distance $d$ on a weighted undirected graph $G$. Our main result is the lower bound $\ell/\Delta(G)\le d$ where $\ell$ is the geodesic distance and $\Delta(G)$ the degree of $G$.…
As well known the rotation distance D(S,T) between two binary trees S, T of n vertices is the minimum number of rotations of pairs of vertices to transform S into T. We introduce the new operation of chain rotation on a tree, involving two…
For $d\ge 2$ and an odd prime power $q$, consider the vector space $\mathbb{F}_q^d$ over the finite field $\mathbb{F}_q$, where the distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined as $\sum_{i=1}^d…
We study the number of distance queries needed to identify certain properties of a hidden tree $T$ on $n$ vertices. A distance query consists of two vertices $x,y$, and the answer is the distance of $x$ and $y$ in $T$. We determine the…
The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many…
The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this…
Let $A$ be an acyclic symmetric matrix of order $n$. There is a weighted forest $F$ whose adjacency matrix is $A$. In this paper, using some results on matching polynomials, we provide an explicit formula for eigenvectors of $A$.