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.