Simplicial effective resistance and enumeration of spanning trees

TL;DR

This paper extends the electrical network analogy to higher-dimensional simplicial complexes, deriving new enumeration formulas for spanning trees and effective resistances, and confirming several conjectures in combinatorics.

Contribution

It introduces a novel approach to compute simplicial effective resistance and enumeration formulas for complex structures, generalizing known results in graph theory.

Findings

Proved a new enumeration formula for color-shifted complexes.

Recovered an enumeration formula for shifted complexes.

Provided explicit expressions for simplicial effective resistances of added facets.

Abstract

A graph can be regarded as an electrical network in which each edge is a resistor. This point of view relates combinatorial quantities, such as the number of spanning trees, to electrical ones such as effective resistance. The second and third authors have extended the combinatorics/electricity analogy to higher dimension and expressed the simplicial analogue of effective resistance as a ratio of weighted tree enumerators. In this paper, we first use that ratio to prove a new enumeration formula for color-shifted complexes, confirming a conjecture by Aalipour and the first author, and generalizing a result of Ehrenborg and van Willigenburg on Ferrers graphs. We then use the same technique to recover an enumeration formula for shifted complexes, first proved by Klivans and the first and fourth authors. In each case, we add facets one at a time, and give explicit expressions for simplicial effective resistances of added facets by constructing high-dimensional analogues of currents and voltages (respectively homological cycles and cohomological cocycles).