The Laplacian on planar graphs and graphs on surfaces

2012/03/06 by Richard Kenyon, Kenyon, Richard · 1 citation
Computer Science · Mathematics · #05C10 #60C #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis #math.CO #math.PR #msc:05C10 #msc:60C

paper · pdf · doi:10.48550/arxiv.1203.1256

arxiv created 2012/03/06 · openalex publication_date 2012/03/06 · arxiv updated 2012/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main goals are to discuss: (1) for "circular" planar networks, the characterization due to Colin de Verdière of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree model; and (3) the characteristic polynomial of the Laplacian on an annulus and torus.

Citations

• Total positivity, Grassmannians, and networks
• Circular planar graphs and resistor networks
• Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird
• Inverse problem in cylindrical electrical networks

Cited by

• Period matrices and homological quasi-trees on discrete Riemann surfaces

Related

• Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions
• The Laplacian Spectra of Graphs and Complex Networks
• Geometric and spectral properties of locally tessellating planar graphs
• A survey of graph laplacians
• On the surface area of graphs, related connectivity measures and spectral estimates