The Elser nuclei sum revisited

TL;DR

This paper revisits Elser's 1984 result on the sum over pandemic edge subsets in a graph, providing a simple proof and exploring its connections to convexity and Morse theory.

Contribution

It offers a straightforward proof of Elser's sum result using sign-reversing involution and discusses its extensions and links to other mathematical areas.

Findings

Proof of Elser's sum result via involution

Extensions and variants of the pandemic subset sum

Connections to convexity and discrete Morse theory

Abstract

Fix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ pandemic if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq \varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and refinements, revealing connections to abstract convexity (the notion of an antimatroid) and discrete Morse theory.