{"paper":{"title":"Sums along the edges of bounded degree graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Georgii Zakharov, Itai Benjamini, Maksim Zhukovskii, Noga Alon","submitted_at":"2025-07-01T18:56:07Z","abstract_excerpt":"Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. What is the minimum size ${\\sf S}_H(G)$ of the set of all sums $A(u)+A(v)$ over all injections $A:V(G)\\to H$? In 2012, the first author, Angel, the second author, and Lubetzky proved that, for expander graphs and $H=\\mathbb{Z}$, this minimum is at least $\\Omega(\\log n)$, and this bound is tight -- there exists a regular expander $G$ with ${\\sf S}_{\\mathbb{Z}}(G)=O(\\log n)$. We prove that, for every constant $d\\geq 3$, the random $d$-regular graph $\\mathcal{G}_{n,d}$ has significantly larger sum-sets: with high probability, for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.01138","kind":"arxiv","version":2},"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/2507.01138/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"}