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In this article, we first introduce the concepts of vector fields and theirdivergence, 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 anextension of Barta Theorem. Then we introduce the mini-max value of a functionon a locally finite and locally connected graph. We show that for a coercivefunction on on a locally finite and locally connected graph, there is amini-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 thelocally finite graph $V$. We show that under suitable assumptions on the graphone has a weak discrete Morse flow for the heat flow on $S$ on any timeinterval. We also study the heat flow with time-variable potential and itsdiscrete Morse flow. We propose the concepts of harmonic maps from a graph to aRiemannian manifold and pose some open questions.

Constants and heat flow on graphs