2020/08/11 by Li Ma, Ma, Li
Computer Science · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Spectral Theory (math.SP) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2008.05025
openalex publication_date 2020/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we first introduce the concepts of vector fields and their divergence, and we recall the concepts of the gradient, Laplacian operator, Cheeger constants, eigenvalues, and heat kernels on a locally finite graph V. We give a projective characteristic of the eigenvalues. We also give an extension of Barta Theorem. Then we introduce the mini-max value of a function on a locally finite and locally connected graph. We show that for a coercive function on on a locally finite and locally connected graph, there is a mini-max value of the function provided it has two strict local minima values. We consider the discrete Morse flow for the heat flow on a finite graph in the locally finite graph V. We show that under suitable assumptions on the graph one has a weak discrete Morse flow for the heat flow on S on any time interval. We also study the heat flow with time-variable potential and its discrete Morse flow. We propose the concepts of harmonic maps from a graph to a Riemannian manifold and pose some open questions.