Geodesic
The Elser nuclei sum revisited
Discrete mathematics & theoretical computer science, Tome 23 (2021-2022) no. 1.

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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.
DOI : 10.46298/dmtcs.7012
Classification : 05E45 
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Grinberg, Darij. The Elser nuclei sum revisited. Discrete mathematics & theoretical computer science, Tome 23 (2021-2022) no. 1. doi : 10.46298/dmtcs.7012. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.7012/

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