Constants and heat flow on graphs

TL;DR

This paper explores fundamental concepts of heat flow, eigenvalues, and vector fields on graphs, extending classical theorems and introducing new variational methods and harmonic map concepts in the graph setting.

Contribution

It introduces a projective characterization of eigenvalues, extends Barta's theorem, and develops discrete Morse flow methods for heat flow on graphs, including variable potential scenarios.

Findings

Eigenvalues characterized projectively.

Extended Barta's theorem for graphs.

Discrete Morse flow for heat flow established.

Abstract

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.