Effective resistance in planar graphs and continued fractions

2025/05/25 by Chan, Swee Hong, Kontorovich, Alex, Pak, Igor
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.19168

Abstract

For a simple graph G=(V,E) and edge e∈ E, the effective resistance is defined as a ratio (τ(G/e))/(τ(G)), where τ(G) denotes the number of spanning trees in G. We resolve the inverse problem for the effective resistance for planar graphs. Namely, we determine (up to a constant) the smallest size of a simple planar graph with a given effective resistance. The results are motivated and closely related to our previous work arXiv:2411.18782 on Sedláček's inverse problem for the number of spanning trees.

Citations

• The spanning tree spectrum: improved bounds and simple proofs
• Equality cases of the Stanley--Yan log-concave matroid inequality
• Euler's idoneal numbers and an inequality concerning minimal graphs with\n a prescribed number of spanning trees
• Partial quotients and representation of rational numbers

Related

• A New Notion of Effective Resistance for Directed Graphs-Part II: Computing Resistances
• Simplicial effective resistance and enumeration of spanning trees
• Algorithmic techniques for finding resistance distances on structured graphs
• Spanning 2-Forests and Resistance Distance in 2-Connected Graphs
• Resistance distance in straight linear 2-trees