Research On Eigenvalue Estimates And Harnack Inequalities Of The P-laplace Operator On Graphs

Eigenvalue estimate and Harnack inequality are classical research topics in stochastic analysis and geometric analysis.In recent years,the geometry and analysis on graphs have attracted the attention of many scholars,how to reasonably define curvature is a primary problem,great progress has been made with the help of Bakry-Emery's square field and the idea of optimal transmission.In this context,this paper studies the eigenvalue estimate of Laplace and p-Laplace operator and Harnack inequality.The specific progress is as follows:(1)Suppose a finite graph satisfying the exponential curvature dimension CDE(m,K)condition,we get the upper bound estimate of the largest eigenvalue of Laplace operator.(2)Inspired by the research method of eigenvalue estimate of the p-Laplace operator on the Riemannian manifold,combined with the characteristics of the graph,we introduced the linearized operator of p-Laplace and the corresponding curvature dimension condition on the finite graph for the first time.Under this condition,we prove the lower bound estimate of the first non-zero eigenvalue and the upper bound estimate of the largest eigenvalue of the p-Laplace operator on the graph,and show that this upper bound estimate is optimal in a sense.(3)Using the linearized p-Laplace operator,we obtain the Harnack inequality and the logarithmic Harnack inequality of the p-Laplace operator on the general finite graphs,which generalizes previous results of Chung-Yau for homogeneous graphs in a sense.