1984/08/01 by Józef Dodziuk · 2 citations
Mathematics · #Stochastic processes and statistical mechanics #Spectral Theory in Mathematical Physics #Markov Chains and Monte Carlo Methods
paper · doi:10.2307/1999107
openalex publication_date 1984/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
The difference Laplacian on a square lattice in \mathbf Rn has been studied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheegerâs bound for the lowest eigenvalue) hold for this difference operator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient.