{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:LLPHPE63A2VYI2QOG45WVRSG2T","short_pith_number":"pith:LLPHPE63","schema_version":"1.0","canonical_sha256":"5ade7793db06ab846a0e373b6ac646d4ffcf1f356d5ac6db342e17fefda8c234","source":{"kind":"arxiv","id":"2607.19817","version":1},"attestation_state":"computed","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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.19817","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-22T06:51:31Z","cross_cats_sorted":[],"title_canon_sha256":"dc4fcea5dc22d71a5b7ecae5b642d7b85e4d7a06ef3c789f0203bd55bcf6d163","abstract_canon_sha256":"7ae598be4264564130c1107a9d5d07ea21d6fcb1184b1a3c94edb1463ccc592d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-23T01:24:00.879829Z","signature_b64":"l1Jc7Fnzi7+ZHQNrUd3lsqRTlEWRzvxiZGhoAVg8xXBhYBfJ2aEEsqLHzTITQjJLajGb+lzZXv3W+U1PtZ7KDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ade7793db06ab846a0e373b6ac646d4ffcf1f356d5ac6db342e17fefda8c234","last_reissued_at":"2026-07-23T01:24:00.878916Z","signature_status":"signed_v1","first_computed_at":"2026-07-23T01:24:00.878916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.19817","created_at":"2026-07-23T01:24:00.879396+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.19817v1","created_at":"2026-07-23T01:24:00.879396+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.19817","created_at":"2026-07-23T01:24:00.879396+00:00"},{"alias_kind":"pith_short_12","alias_value":"LLPHPE63A2VY","created_at":"2026-07-23T01:24:00.879396+00:00"},{"alias_kind":"pith_short_16","alias_value":"LLPHPE63A2VYI2QO","created_at":"2026-07-23T01:24:00.879396+00:00"},{"alias_kind":"pith_short_8","alias_value":"LLPHPE63","created_at":"2026-07-23T01:24:00.879396+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.04367","citing_title":"Extremal Graphs for the Energy-Independence Number Inequality","ref_index":8,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LLPHPE63A2VYI2QOG45WVRSG2T","json":"https://pith.science/pith/LLPHPE63A2VYI2QOG45WVRSG2T.json","graph_json":"https://pith.science/api/pith-number/LLPHPE63A2VYI2QOG45WVRSG2T/graph.json","events_json":"https://pith.science/api/pith-number/LLPHPE63A2VYI2QOG45WVRSG2T/events.json","paper":"https://pith.science/paper/LLPHPE63"},"agent_actions":{"view_html":"https://pith.science/pith/LLPHPE63A2VYI2QOG45WVRSG2T","download_json":"https://pith.science/pith/LLPHPE63A2VYI2QOG45WVRSG2T.json","view_paper":"https://pith.science/paper/LLPHPE63","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.19817&json=true","fetch_graph":"https://pith.science/api/pith-number/LLPHPE63A2VYI2QOG45WVRSG2T/graph.json","fetch_events":"https://pith.science/api/pith-number/LLPHPE63A2VYI2QOG45WVRSG2T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LLPHPE63A2VYI2QOG45WVRSG2T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LLPHPE63A2VYI2QOG45WVRSG2T/action/storage_attestation","attest_author":"https://pith.science/pith/LLPHPE63A2VYI2QOG45WVRSG2T/action/author_attestation","sign_citation":"https://pith.science/pith/LLPHPE63A2VYI2QOG45WVRSG2T/action/citation_signature","submit_replication":"https://pith.science/pith/LLPHPE63A2VYI2QOG45WVRSG2T/action/replication_record"}},"created_at":"2026-07-23T01:24:00.879396+00:00","updated_at":"2026-07-23T01:24:00.879396+00:00"}