{"paper":{"title":"Energy and independence number","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hitesh Kumar, Shivaramakrishna Pragada","submitted_at":"2026-07-22T06:51:31Z","abstract_excerpt":"For a graph $G$ of order $n$, with adjacency eigenvalues $\\lambda_1(G) \\geq \\cdots \\geq \\lambda_n(G)$, the \\emph{energy} of $G$ is defined to be \\[\\mathcal{E}(G)=\\sum_{i=1}^{n} |\\lambda_i(G)|.\\] A well-known conjecture from the 1980s by Fajtlowicz states that for any graph $G$, \\[\\mathcal{E}(G) \\ge 2\\left(n-\\alpha(G)\\right),\\] where $\\alpha(G)$ denotes the independence number. We prove this conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19817","kind":"arxiv","version":1},"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/2607.19817/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"}