Effective resistance in planar graphs and continued fractions

TL;DR

This paper characterizes the minimal size of simple planar graphs with a specified effective resistance, linking graph structure to spanning tree counts and resolving an inverse problem in graph theory.

Contribution

It provides a solution to the inverse problem for effective resistance in planar graphs, determining minimal graph sizes for given resistance values.

Findings

Resolved the inverse problem for effective resistance in planar graphs.

Identified the minimal size of planar graphs with a specified effective resistance.

Connected effective resistance to the number of spanning trees in planar graphs.

Abstract

For a simple graph $G=(V,E)$ and edge $e\in E$, the effective resistance is defined as a ratio $\frac{\tau(G/e)}{\tau(G)}$, where $\tau(G)$ 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 Sedl\'a\v{c}ek's inverse problem for the number of spanning trees.