The main focus in this memoir is on Laplacians on both weighted graphs and weighted metric graphs. The authors consider infinite locally finite graphs and do not make any further geometric assumptions. Although the existing literature usually treats these two types of Laplacian operators separately, in this book the authors approach them in a uniform manner and emphasiize the relationship between them. One of this book's main conceptual messages is that these two settings should be regarded as complementary (rather than opposite) and their interplay leads to further important insight on both sides.

The authors' central goal is twofold. First, they explore the relationships between these two objects by comparing their basic spectral (self-adjointness, spectral gap, etc.), parabolic (Markovian uniqueness, recurrence, stochastic completeness, etc.), and metric (quasi isometries, intrinsic metrics, etc.) properties. In turn, the authors employ these connections either to prove new results for Laplacians on metric graphs or to provide new proofs and perspective on the recent progress in weighted graph Laplacians. The authors also demonstrate their findings by considering several important classes of graphs (Cayley graphs, tessellations, and antitrees).