On the Lichnerowicz estimate for Laplacians on graphs under spectral curvature assumptions
type: Item
instance of: scholarly article
title: On the Lichnerowicz estimate for Laplacians on graphs under spectral curvature assumptions
zbMATH Open document ID: 1504.05165
DOI: 10.1134/S0001434622090164
author: Yanyan Li
published in: Mathematical Notes
publication date: 2022-10-25T00:00:00Z
review text: A triple \(G=(V,w,m)\) that consists of a finite vertex set \(V\), a symmetric function \(w: V\times V\rightarrow [0,\infty)\) which is zero on diagonal, and a vertex measure \(m: V\rightarrow (0,\infty)\) is called a finite weighted and measured graph. Let \(C(V)\) denote the set of real valued functions defined on \(V\). On the graph \(G\), the discrete Laplacian \(\Delta: C(V)\rightarrow C(V)\) is given in the following way \[ \Delta f(x):=\sum_{y\in V} \frac{w(x,y)}{m(x)} (f(y)-f(x)),\quad x\in V. \] Let \(K>0\), \(n\in(0,\infty]\) and \(\rho\in C(V)\). It is said that the \((K,n,\rho)\)-spectrally positive Ricci curvature dimension condition on \(G\) is satisfied if and only if the following two conditions are satisfied: \par i) the variable curvature dimension condition \(CD(\rho,n)\) holds \[ \Gamma_2(f)(x)\geq \rho(x)\Gamma(f)(x)+\frac{1}{n}(\Delta f)^2(x),\quad f\in C(V), \quad x\in V. \] ii) as well as \[ -\frac{1}{2}\langle -\Delta f,f\rangle+\langle \rho f,f\rangle\geq K\langle f,f\rangle,\quad f\in C(V). \] The following eigenvalue estimate is established and proved. Let \(K>0\), \(n\in(1,\infty]\) and \(\rho\in C(V)\). Let \(G=(V,w,m)\) be a finite weighted and measured graph on which the \((K,n,\rho)\)-spectrally positive Ricci curvature dimension condition is satisfied. Then, \[ \lambda\geq \frac{n}{n-1}K, \] where \(\lambda=\lambda(G)\) is the first positive eigenvalue of \(\Delta\) in \(\ell^2(V,m)\).
reviewed by: Milica Andelić
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C22
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 53B21
zbMATH DE Number: 7606794
zbMATH Keywords: finite weighted and measured graph
zbMATH Keywords: Ricci curvature dimension
zbMATH Keywords: eigenvalue estimate
MaRDI profile type: Publication
cites work: Q5186515
cites work: Geometric and spectral estimates based on spectral Ricci curvature assumptions
cites work: Spectrally positive Bakry-Émery Ricci curvature on graphs
cites work: Curvature aspects of graphs
description: scientific article; zbMATH DE number 7606794