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We prove that, for every real number $p\\ge 2$ and every simple connected graph $G$ on $n$ vertices, $$ \\mathcal E_p(G)\\ge \\mathcal E_p(P_n), $$ where $P_n$ denotes the path on $n$ vertices. Moreover, for each fixed $p>2$, equality holds if and only if $G\\cong P_n$. Together with the previously known star-minimality results, this completes the sol"},"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":"2605.22730","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-21T17:03:10Z","cross_cats_sorted":[],"title_canon_sha256":"95127cc3b6ec56cfedecd54a0b4c063b39d012a99be99afd189c691a71f809cc","abstract_canon_sha256":"d8638eae7f862351b7c487f227d8bc095468fbf0ee1823097ad5963cb9b56ecf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T02:04:52.323424Z","signature_b64":"i+MXQjUPKK556rqM479cxnxE4vLsAHreTMLReaetckcfldMVCwNanhHmtToNBm9I0b0pe7lfao37OOCK9UcIDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd8d958186e41159323e4231f738d32790506a86db43b9012d4e07e28745dca8","last_reissued_at":"2026-05-22T02:04:52.322760Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T02:04:52.322760Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Path-Minimality of $p$-Energy for Connected Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Quanyu Tang, Yinchen Liu","submitted_at":"2026-05-21T17:03:10Z","abstract_excerpt":"Let $G$ be a simple connected graph on $n$ vertices, and let $\\lambda_1(G),\\lambda_2(G),\\ldots,\\lambda_n(G)$ be the eigenvalues of its adjacency matrix $A(G)$. For $p>0$, define the $p$-energy of $G$ by $\\mathcal E_p(G)=\\sum_{i=1}^n |\\lambda_i(G)|^p$. We prove that, for every real number $p\\ge 2$ and every simple connected graph $G$ on $n$ vertices, $$ \\mathcal E_p(G)\\ge \\mathcal E_p(P_n), $$ where $P_n$ denotes the path on $n$ vertices. Moreover, for each fixed $p>2$, equality holds if and only if $G\\cong P_n$. Together with the previously known star-minimality results, this completes the sol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22730","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/2605.22730/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":"2605.22730","created_at":"2026-05-22T02:04:52.322875+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.22730v1","created_at":"2026-05-22T02:04:52.322875+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22730","created_at":"2026-05-22T02:04:52.322875+00:00"},{"alias_kind":"pith_short_12","alias_value":"XWGZLAMG4QIV","created_at":"2026-05-22T02:04:52.322875+00:00"},{"alias_kind":"pith_short_16","alias_value":"XWGZLAMG4QIVSMR6","created_at":"2026-05-22T02:04:52.322875+00:00"},{"alias_kind":"pith_short_8","alias_value":"XWGZLAMG","created_at":"2026-05-22T02:04:52.322875+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XWGZLAMG4QIVSMR6IIY7OOGTE6","json":"https://pith.science/pith/XWGZLAMG4QIVSMR6IIY7OOGTE6.json","graph_json":"https://pith.science/api/pith-number/XWGZLAMG4QIVSMR6IIY7OOGTE6/graph.json","events_json":"https://pith.science/api/pith-number/XWGZLAMG4QIVSMR6IIY7OOGTE6/events.json","paper":"https://pith.science/paper/XWGZLAMG"},"agent_actions":{"view_html":"https://pith.science/pith/XWGZLAMG4QIVSMR6IIY7OOGTE6","download_json":"https://pith.science/pith/XWGZLAMG4QIVSMR6IIY7OOGTE6.json","view_paper":"https://pith.science/paper/XWGZLAMG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.22730&json=true","fetch_graph":"https://pith.science/api/pith-number/XWGZLAMG4QIVSMR6IIY7OOGTE6/graph.json","fetch_events":"https://pith.science/api/pith-number/XWGZLAMG4QIVSMR6IIY7OOGTE6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XWGZLAMG4QIVSMR6IIY7OOGTE6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XWGZLAMG4QIVSMR6IIY7OOGTE6/action/storage_attestation","attest_author":"https://pith.science/pith/XWGZLAMG4QIVSMR6IIY7OOGTE6/action/author_attestation","sign_citation":"https://pith.science/pith/XWGZLAMG4QIVSMR6IIY7OOGTE6/action/citation_signature","submit_replication":"https://pith.science/pith/XWGZLAMG4QIVSMR6IIY7OOGTE6/action/replication_record"}},"created_at":"2026-05-22T02:04:52.322875+00:00","updated_at":"2026-05-22T02:04:52.322875+00:00"}