Related papers: Distance matrices perturbed by a Laplacian

Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$…

Let $M=[m_{ij}]$ be an $n\times m$ real matrix, $\rho$ be a nonzero real number, and $A$ be a symmetric real matrix. We denote by $D(M)$ the $n\times n$ diagonal matrix $diag(\sum_{j=1}^{m}m_{1j},\ldots,\sum_{j=1}^{m}m_{nj})$ and denote by…

A matrix $S=(s_{ij})\in{\mathbb R}^{n\times n}$ is said to determine a \emph{transitional measure} for a digraph $G$ on $n$ vertices if for all $i,j,k\in\{1,\...,n\},$ the \emph{transition inequality} $s_{ij} s_{jk}\le s_{ik} s_{jj}$ holds…

For a graph $G=(V, E)$ and $i, j\in V$, denote the distance between $i$ and $j$ in $G$ by $D(i, j)$ and the degrees of $i$, $j$ by $d_i$, $d_j$, respectively. Let $f(D(i, j), d_{i}, d_{j})$ be a function symmetric in $i$ and $j$. Define a…

For a graph $G=(V,E)$, assigning each edge $e\in E$ a weight of a dual number $w(e)=1+\widehat{a}_{e}\varepsilon$, the weighted graph $G^{w}=(V,E,w)$ is called a dual number weighted graph, where $-\widehat{a}_{e}$ can be regarded as the…

This work explores the definiteness of the weighted graph Laplacian matrix with negative edge weights. The definiteness of the weighted Laplacian is studied in terms of certain matrices that are related via congruent and similarity…

A new class of distances for graph vertices is proposed. This class contains parametric families of distances which reduce to the shortest-path, weighted shortest-path, and the resistance distances at the limiting values of the family…

Let $G$ be a connected graph on $n$ vertices, and let $D(G)$ be the distance matrix of $G$. Let $\partial_1(G)\ge\partial_2(G)\ge\cdots\ge\partial_n(G)$ denote the eigenvalues of $D(G)$. In this paper, we characterize all connected graphs…

For an arbitrary set of distances $D\subseteq \{0,1, \ldots, diam(G)\}$, a $D$-weight of a vertex $x$ in a graph $G$ under a vertex labeling $f:V\rightarrow \{1,2, \ldots , v\}$ is defined as $w_D(x)=\sum_{y\in N_D(x)} f(y)$, where $N_D(x)…

Let $D$ be an digraph of order $n$ with adjacency matrix $A(D)$ and outdegree matrix $\Delta^+=\Delta^+(D)$. Then the Laplacian and signless Laplacian matrices of $D$ are respectively defined as $L(D)=\Delta^+-A(D)$ and…

Let $G$ be a simple undirected connected graph with the Harary matrix $RD(G)$, which is also called the reciprocal distance matrix of $G$. The reciprocal distance signless Laplacian matrix of $G$ is $RQ(G)=RT(G)+RD(G)$, where $RT(G)$…

The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been…

Given a connected graph $G\ $of order $n$ and a nonnegative symmetric matrix $A=\left[ a_{i,j}\right] $ of order $n,$ define the function $F_{A}\left( G\right) $ as% \[ F_{A}\left( G\right) =\sum_{1\leq i<j\leq n}d_{G}\left( i,j\right)…

For a graph \(G\) with no isolated vertices, its Laplacian ratio is defined as \[ \pi(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] where \(L(G)\) is the Laplacian matrix of \(G\), \(d(v)\) is the degree of \(v\), and…

Suppose that $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d( v_i,v_j ) $ be the distance between $v_i$ and $v_j$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} )_{n\times n}$, where…

Let $k$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $\delta$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge…

The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the…

Let $T$ be a tree with a vertex set $\{ 1,2,\dots, N \}$. Denote by $d_{ij}$ the distance between vertices $i$ and $j$. In this paper, we present an explicit combinatorial formula of principal minors of the matrix $(t^{d_{ij}})$, and its…

Let $G$ be a connected graph with vertex set $\{v_1, v_2, \ldots, v_\mathbf{n}\}$. As a variant of the classical distance matrix, the \emph{exponential distance matrix} was introduced independently by Yan and Yeh, and by Bapat et al. For a…

A signed graph is a graph whose edges are labeled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize balance in signed…