Related papers: Squared distance matrices of trees with matrix wei…

Given a rooted tree $T$ with vertices $u_1,u_2,\ldots,u_n$, the level matrix $L(T)$ of $T$ is the $n \times n$ matrix for which the $(i,j)$-th entry is the absolute difference of the distances from the root to $v_i$ and $v_j$. This matrix…

Let $n \geq 4$ be an even integer and $W_n$ be the wheel graph with $n$ vertices. The distance $d_{ij}$ between any two distinct vertices $i$ and $j$ of $W_n$ is the length of the shortest path connecting $i$ and $j$. Let $D$ be the $n…

A result of Bapat and Sivasubramanian gives the inertia of the distance squared matrix of a tree. We develop general tools on how pendant vertices and degree 2 vertices affect the inertia of the distance squared matrix and use these to give…

This paper is concerned with the distance of a symmetric tridiagonal Toeplitz matrix $T$ to the variety of similarly structured singular matrices, and with determining the closest matrix to $T$ in this variety. Explicit formulas are…

Let $G=(V_G, E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $\xi^{d}(G) = \sum_{v\in V_G}\varepsilon_{G}(v)D_{G}(v)$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v) =…

Let $W_n$ denote the wheel graph having $n$-vertices. If $i$ and $j$ are any two vertices of $W_n$, define \[d_{ij}:= \begin{cases} 0 & \mbox{if}~i=j \\ 1 & \mbox{if}~i~ \mbox{and} ~j~ \mbox{are adjacent} \\ 2 & \mbox{else}. \end{cases}\]…

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}…

Let $G=(V,E)$ be a simple, unweighted, connected graph. Let $d(u,v)$ denote the distance between vertices $u,v$. A resolving set of $G$ is a subset $S$ of $V$ such that knowing the distance from a vertex $v$ to every vertex in $S$ uniquely…

We consider three different schemes for signal routing on a tree. The vertices of the tree represent transceivers that can transmit and receive signals, and are equipped with i.i.d. weights representing the strength of the transceivers. The…

Distance well-defined graphs consist of connected undirected graphs, strongly connected directed graphs and strongly connected mixed graphs. Let $G$ be a distance well-defined graph, and let ${\sf D}(G)$ be the distance matrix of $G$.…

Let $T = ([n], E)$ be a tree and let $D = ( d(i,j) )_{i, j \le n}$ be the distance matrix of $T$. Let $S\subseteq [n]$. We give the first combinatorial proof for a formula to compute the principal minor of $D$ indexed by $S$, namely $\det…

Let $T$ be a weighted tree. The weight of a subtree $T_1$ of $T$ is defined as the product of weights of vertices and edges of $T_1$. We obtain a linear-time algorithm to count the sum of weights of subtrees of $T$. As applications, we…

A (pseudo-)metric $D$ on a finite set $X$ is said to be a `tree metric' if there is a finite tree with leaf set $X$ and non-negative edge weights so that, for all $x,y \in X$, $D(x,y)$ is the path distance in the tree between $x$ and $y$.…

A connected graph is called a multi-block graph if each of its blocks is a complete multi-partite graph. Building on the work of \cite{Bp3,Hou3}, we compute the determinant and inverse of the distance matrix for a class of multi-block…

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…

In this paper, we prove two theorems concerning the sums of squared distances between points on a unit $n$-sphere that generalize two facts previously known about the case where the points are the vertices of a regular polygon. The first…

It is an open question whether there exists a polynomial-time algorithm for computing the rotation distances between pairs of extended ordered binary trees. The problem of computing the rotation distance between an arbitrary pair of trees,…

It is known from the algebraic graph theory that if $L$ is the Laplacian matrix of some tree $G$ with a vertex degree sequence $\mathbf{d}=(d_1, ..., d_n)^\top$ and $D$ is its distance matrix, then…

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 T be a weighted tree with n leaves. Let D_{i,j} be the distance between the leaves i and j. Let D_{i,j,k}= (D_{i,j} + D_{j,k} +D_{i,k})/2. We will call such numbers "triple weights" of the tree. In this paper, we give a…