Incidence and Equilibrium on Graphs: How Conservation Leads to Integrality, Laplacians, Resistor Networks, and Random Walks
An expository tour of the incidence matrix as a unifying operator: conservation laws, total unimodularity, integral network optimization, graph Laplacians, resistor networks, harmonic functions, and random walks. Published technical note.
Abstract
A surprising amount of graph theory and applied mathematics flows from one small object: the incidence matrix. This expository note follows it through conservation laws, total unimodularity, and integral network optimization, then on to graph Laplacians, resistor networks, harmonic functions, and random walks. The recurring theme is that conservation, integrality, equilibrium, and averaging are not separate stories — they are different views of the same underlying operator structure.