Related papers: Graph curvature via resistance distance

We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by $K>0$ have diameter bounded by $\mbox{diam}(G) \leq…

Let $G$ be a connected graph with $n$ vertices. The resistance distance $\Omega_{G}(i,j)$ between any two vertices $i$ and $j$ of $G$ is defined as the effective resistance between them in the electrical network constructed from $G$ by…

Resistance distance has been studied extensively in the past years, with the majority of previous studies devoted to undirected networks, in spite of the fact that various realistic networks are directed. Although several generalizations of…

Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency…

Steinerberger proposed a notion of curvature on graphs involving the graph distance matrix (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and…

We study the linearization of a discrete transportation distance between probability distributions on finite weighted graphs originally due to Maas (``Gradient flows of the entropy for finite {M}arkov chains,'' J. Funct. Anal. 261(8), 2011)…

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably,…

We prove distance bounds for graphs possessing positive Bakry-\'Emery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits…

We introduce a natural notion of mean (or average) distance in the context of compact metric graphs, and study its relation to geometric properties of the graph. We show that it exhibits a striking number of parallels to the reciprocal of…

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…

Let $G$ be a strongly connected and balanced directed graph. The Laplacian matrix of $G$ is then the matrix (not necessarily symmetric) $L:=D-A$, where $A$ is the adjacency matrix of $G$ and $D$ is the diagonal matrix such that the row sums…

We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph. This curvature, termed…

In graph signal processing, data samples are associated to vertices on a graph, while edge weights represent similarities between those samples. We propose a convex optimization problem to learn sparse well connected graphs from data. We…

Given a finite, simple, connected graph $G=(V,E)$ with $|V|=n$, we consider the associated graph Laplacian matrix $L = D - A$ with eigenvalues $0 = \lambda_1 < \lambda_2 \leq \dots \leq \lambda_n$. One can also consider the same graph…

Density matrices of graphs are combinatorial laplacians normalized to have trace one (Braunstein \emph{et al.} \emph{Phys. Rev. A,} \textbf{73}:1, 012320 (2006)). If the vertices of a graph are arranged as an array, then its density matrix…

The first non-zero Laplacian eigenvalue $\lambda_1$ of a finite graph is bounded below by its minimum Lin--Lu--Yau curvature $\kappa$. This is a discrete analogue of the classical Lichnerowicz Theorem. A graph with $\lambda_1=\kappa$ is…

In this paper we provide the group inverse of the combinatorial Laplacian matrix of distance-biregular graphs using the so-called equilibrium measures for sets obtained by deleting a vertex. We also show that the two equilibrium arrays…

We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…

In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph. A weighted graph $G=(V,E,d)$ is an…

We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and…