2022/06/05 by Art M. Duval, Duval, Art M., Woong Kook +5
Computer Science · Mathematics · #05C50 #31C20 #94C15 #Combinatorics (math.CO) #Computational Drug Discovery Methods #FOS: Mathematics #Graph theory and applications #Primary 05E45 #Secondary 05C05 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2206.02182
openalex publication_date 2022/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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).