Metric properties of electrical networks and the graph reconstruction problems

2026/01/13 by V. G. Gorbounov, A. A. Kazakov · 1 voice
Mathematics · #math.CO

paper · pdf

Abstract

Using the generalized Temperley trick, we demonstrate the explicit embedding of circular electrical networks into totally non-negative Grassmannians. Building on this result, we show that the effective resistances between boundary nodes of circular electrical networks satisfy the Kalmanson property, and we provide the full characterization of planar electrical Kalmanson metrics. Additionally, we present a graph reconstruction algorithm with applications in phylogenetic network analysis as well as the numerical solution of the Calderon problem.

Citations

• Inverse problems related to electrical networks and the geometry of non-negative Grassmannians
• Circular planar electrical networks, Split systems, and Phylogenetic networks
• Phylogenetic networks as circuits with resistance distance
• Electrical varieties as vertex integrable statistical models
• Electroid varieties and a compactification of the space of electrical networks
• The Laplacian on planar graphs and graphs on surfaces
• Geometry of Cuts and Metrics
• Electrical impedance tomography and Calderón's problem
• Electrical impedance tomography with resistor networks

Discussions

• Metric properties of electrical networks and the graph reconstruction problems https://arxiv.org/abs/2601.08465v1 [bsky, 0 points, 0 comments]

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• Circular planar electrical networks, Split systems, and Phylogenetic networks
• Electrical networks and data analysis in phylogenetics
• Inverse problems related to electrical networks and the geometry of non-negative Grassmannians
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• Recovering Conductances of Resistor Networks in a Punctured Disk