Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.12048

An Introductory Survey of Recursions in the Computation of Resistance Distance

Emily J. Evans·Russell Jay Hendel
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper presents an introduction and expository account of a beautiful, current, and active application of recursions to the computation of resistance distance. Resistance distance, also referred to as effective resistance, is a well-known graph metric that arises naturally by considering a graph as an electrical circuit; heuristically resistance distance measures both the number of paths between two vertices in a graph and the cost of each path. This topic finds applications in a rich array of fields including social, biological, ecological, and transportation networks, chemistry, graph theory, numerical linear algebra, and engineering. A variety of methods are used in the field to determine resistance distance including recursive, mathematical, and graphical techniques. Sequences familiar to the readers of the Fibonacci Quarterly such as the Fibonacci and Lucas sequences appear quite often in results in the literature. Twenty five to forty years ago there were a handful of papers on resistance that appeared in the Fibonacci Quarterly and the Proceedings and recently papers on the subject have appeared again. It is hoped that this introductory expository account will interest readers of the Quarterly to renew interest in this current and active field.

Explore related subjects

math.HOmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Emily J. Evans, Russell Jay Hendel. 2025-05-26. An Introductory Survey of Recursions in the Computation of Resistance Distance. https://arxiv.org/abs/2506.12048

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Stairs of Reconciliation: A Mathematical Tourist in Graz

Inside the Grazer Burg, two late-Gothic stone flights rise about distinct spindles, overlap, share several treads, and separate again. Their plan is governed not by a coaxial double helix but, to first approximation, by two intersecting circles. This elementary geometry yields a model of recurrent meeting and makes explicit the compatibility conditions that meeting requires. It also leads to a second object that geometers call a double spiral staircase - the helicoid - and to a useful distinction between resemblance and identity. The staircase becomes a meditation on how paths, models, and disciplines can meet without becoming the same.

math.HO↗
arXiv2026

On the Reconstruction of SAS from Other Triangle Congruence Criteria

Starting from a Hilbert plane and removing the Side-Angle-Side (SAS) congruence axiom, we investigate to what extent SAS can be recovered synthetically from the remaining classical triangle congruence criteria. We show that the Angle-Side-Angle criterion, together with a ray correspondence principle corresponding to Theorem 13 of Hilbert's \emph{Grundlagen der Geometrie}, suffices to reconstruct SAS. We further show that both the Side-Side-Side and the Side-Angle-Angle criteria also suffice, once combined with the ray correspondence principle and suitable auxiliary principles -- the existence of midpoints and a hypotenuse-angle criterion for right triangles in the first case, and the existence of angle bisectors, the congruence of supplements of congruent angles, and the Pons Asinorum in the second. Although the two routes rely on auxiliary principles of different character, we show that they converge on a single final argument once a common hypotenuse-angle criterion is established. A metamathematical analysis, based on an explicit model adapted from Hilbert's own independence construction, complements these reconstructions: it shows that the ray correspondence principle alone cannot reconstruct any of the classical criteria, and that the Pons Asinorum and the hypotenuse-angle criterion are each independent of the remaining auxiliary principles used in their respective reconstructions. The resulting picture is not a formal hierarchy of the congruence criteria, but it does show that the Angle-Side-Angle reconstruction rests on a provably more economical basis than those obtained from Side-Side-Side or Side-Angle-Angle.

math.HO↗
arXiv2026

Come for the vibe, stay for the math

This article describes our experiences in mathematical outreach over the past decade. We talk about specific activities, but also general principles that we've learned along the way.

math.HO↗