Algorithmic techniques for finding resistance distances on structured graphs

TL;DR

This paper surveys various methods for calculating resistance distances in structured graphs, highlighting their applications in science and engineering, and discussing both exact and approximate techniques with illustrative examples.

Contribution

It provides a comprehensive overview of existing methods for resistance distance calculation, including new open questions and conjectures in the field.

Findings

Various exact and approximate methods are summarized.

Illustrative examples demonstrate each technique.

Open questions suggest directions for future research.

Abstract

In this paper we give a survey of methods used to calculate values of resistance distance (also known as effective resistance) in graphs. Resistance distance has played a prominent role not only in circuit theory and chemistry, but also in combinatorial matrix theory and spectral graph theory. Moreover resistance distance has applications ranging from quantifying biological structures, distributed control systems, network analysis, and power grid systems. In this paper we discuss both exact techniques and approximate techniques and for each method discussed we provide an illustrative example of the technique. We also present some open questions and conjectures.