The Elser Nuclei Sum Revisited

Abstract

Fix a finite undirected graph Γ and a vertex v of Γ. Let E be the set of edges of Γ. We call a subset F of E pandemic if each edge of Γ 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 (−1)|F| over all pandemic subsets F of E is 0 if E≠∅. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and a refinement using discrete Morse theory.