Probability, Curvature and Spectrum on Graphs

We show that, for a metric graph equipped with the Laplacian operator $\mathrm{\Delta }=-\frac{d^2}{d{x}^2}$, the graph trace formula admits a new interpretation in terms of quantum probability and curvature. Our approach is based on a notion of graph curvature inspired by Wilson-lines and holonomy, together with von-Neumann’s ergodic theorem.