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The norm $\\left\\Vert G\\right\\Vert _{\\ast}$ has been intensively studied under the name of graph energy, a concept introduced by Gutman in 1978.\n  This note studies the maximum trace norm of $r$-partite graphs, which raises some unusual problems for $r>2$. It is shown that, if $G$ is an $r$-partite graph of order $n,$ then \\[ \\left\\Vert G\\right\\Vert _{\\ast}<\\frac{n^{3/2}}{2}\\sqrt{1-1/r}+\\left( 1-1/r\\right) n. \\] For some special $r$ this bound is tight"},"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":"1502.04342","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-15T18:12:23Z","cross_cats_sorted":[],"title_canon_sha256":"4943624f9128a9c478f19d0b84ed1bcae776c1af9308f76d91cb6164defca8ad","abstract_canon_sha256":"e9f5bf96b369ba5227785b0e407276dabf6594a6f0b36c33002b523b5faeab17"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:02.742639Z","signature_b64":"3KivtXoqDCWuuyo/hsYi313auu03tnjKPws0r208s9KVFQjpsllXQOAr+0EbS1mRe/boTz8rbJHDFfgyk1nIDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e4af9811a467e231ad50e3ba8c2123431f57a43374957195136b8a6e55f05c27","last_reissued_at":"2026-05-18T02:20:02.741815Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:02.741815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The trace norm of r-partite graphs and matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"V. 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It is shown that, if $G$ is an $r$-partite graph of order $n,$ then \\[ \\left\\Vert G\\right\\Vert _{\\ast}<\\frac{n^{3/2}}{2}\\sqrt{1-1/r}+\\left( 1-1/r\\right) n. \\] For some special $r$ this bound is tight"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04342","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":""},"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":"1502.04342","created_at":"2026-05-18T02:20:02.741945+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.04342v2","created_at":"2026-05-18T02:20:02.741945+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.04342","created_at":"2026-05-18T02:20:02.741945+00:00"},{"alias_kind":"pith_short_12","alias_value":"4SXZQENEM7RD","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"4SXZQENEM7RDDLKQ","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"4SXZQENE","created_at":"2026-05-18T12:29:05.191682+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/4SXZQENEM7RDDLKQ4O5IYIJDIM","json":"https://pith.science/pith/4SXZQENEM7RDDLKQ4O5IYIJDIM.json","graph_json":"https://pith.science/api/pith-number/4SXZQENEM7RDDLKQ4O5IYIJDIM/graph.json","events_json":"https://pith.science/api/pith-number/4SXZQENEM7RDDLKQ4O5IYIJDIM/events.json","paper":"https://pith.science/paper/4SXZQENE"},"agent_actions":{"view_html":"https://pith.science/pith/4SXZQENEM7RDDLKQ4O5IYIJDIM","download_json":"https://pith.science/pith/4SXZQENEM7RDDLKQ4O5IYIJDIM.json","view_paper":"https://pith.science/paper/4SXZQENE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.04342&json=true","fetch_graph":"https://pith.science/api/pith-number/4SXZQENEM7RDDLKQ4O5IYIJDIM/graph.json","fetch_events":"https://pith.science/api/pith-number/4SXZQENEM7RDDLKQ4O5IYIJDIM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4SXZQENEM7RDDLKQ4O5IYIJDIM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4SXZQENEM7RDDLKQ4O5IYIJDIM/action/storage_attestation","attest_author":"https://pith.science/pith/4SXZQENEM7RDDLKQ4O5IYIJDIM/action/author_attestation","sign_citation":"https://pith.science/pith/4SXZQENEM7RDDLKQ4O5IYIJDIM/action/citation_signature","submit_replication":"https://pith.science/pith/4SXZQENEM7RDDLKQ4O5IYIJDIM/action/replication_record"}},"created_at":"2026-05-18T02:20:02.741945+00:00","updated_at":"2026-05-18T02:20:02.741945+00:00"}