2007/12/10 by Radosław K. Wojciechowski, Wojciechowski, Radoslaw K. · 3 citations
Computer Science · Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Graph theory and applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.0712.1570
openalex publication_date 2007/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this thesis, we analyze the stochastic completeness of a heat kernel on graphs which is a function of three variables: a pair of vertices and a continuous time, for infinite, locally finite, connected graphs. For general graphs, a sufficient condition for stochastic completeness is given in terms of the maximum valence on spheres about a fixed vertex. That this result is optimal is shown by studying a particular family of trees. We also prove a lower bound on the bottom of the spectrum for the discrete Laplacian and use this lower bound to show that in certain cases the Laplacian has empty essential spectrum.