Harnack's inequality and Green's functions on locally finite graphs

In this paper, we study the gradient estimate for positive solutions of Schrodinger equations on locally finite and connected graphs. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations on such graphs. We also set up some existence results about Green's functions of the Laplacian equations on locally finite graphs. We derive a lower bound for the principal eigenvalue of the Laplace operator in terms of the upper bound of total integral of Green's function. Interesting existence results for positive solutions of Schrodinger equations are derived via the use of related principal eigenvalues.