{"paper":{"title":"The Elser nuclei sum revisited","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Darij Grinberg","submitted_at":"2020-09-24T07:36:30Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.11527","kind":"arxiv","version":10},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2009.11527/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}