{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:RTTGLDSE6UPPUZXGP5ZA24MBIE","short_pith_number":"pith:RTTGLDSE","schema_version":"1.0","canonical_sha256":"8ce6658e44f51efa66e67f720d71814122fce448926eba40aa2804264b646a5b","source":{"kind":"arxiv","id":"2605.13733","version":1},"attestation_state":"computed","paper":{"title":"Helmholzian Spectra of Graphs: Novel Properties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A new graph-theoretic proof confirms that the Helmholtzian matrix represents the graph Helmholtzian operator, classifying graphs with exactly two distinct eigenvalues and giving combinatorial meaning to its polynomial coefficients.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jianfeng Wang, Lu Lu, Yi Wang, Yongtang Shi, Zoran Stani\\'c","submitted_at":"2026-05-13T16:12:19Z","abstract_excerpt":"Let $\\grad$, $\\curl$, and $\\dv$ be the graph-theoretic analogues of the gradient, curl, and divergence operators from multivariate calculus. The graph Laplacian $-\\dv \\grad$ gives rise to the celebrated Laplacian matrix, while the matrix representation of the graph Helmholtzian $\\grad \\grad^* + \\curl^* \\curl$ is called the Helmholtzian matrix. In this paper, we present a new graph-theoretic proof that the Helmholtzian matrix indeed represents the graph Helmholtzian. We then investigate the spectral properties of this matrix. Our main results are as follows: (i) a classification of graphs havin"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.13733","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T16:12:19Z","cross_cats_sorted":[],"title_canon_sha256":"13f9cd976c5d8964c2ccfce0a5dd75a0e83c794891a01d3214326feab0a0c6ad","abstract_canon_sha256":"e059a85da1893002ec477a7089a18840f938d413e2871a03c4fe1129856274a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:16.540061Z","signature_b64":"PTObkXcZgWD3vKz2HyuzIRxSMgb3dxB9fiLoPO6JvD/xk3hSu8wNLbjQRJJIfQh6MImsfZuZF6k4XZT18HWrCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ce6658e44f51efa66e67f720d71814122fce448926eba40aa2804264b646a5b","last_reissued_at":"2026-05-18T02:44:16.539635Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:16.539635Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Helmholzian Spectra of Graphs: Novel Properties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A new graph-theoretic proof confirms that the Helmholtzian matrix represents the graph Helmholtzian operator, classifying graphs with exactly two distinct eigenvalues and giving combinatorial meaning to its polynomial coefficients.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jianfeng Wang, Lu Lu, Yi Wang, Yongtang Shi, Zoran Stani\\'c","submitted_at":"2026-05-13T16:12:19Z","abstract_excerpt":"Let $\\grad$, $\\curl$, and $\\dv$ be the graph-theoretic analogues of the gradient, curl, and divergence operators from multivariate calculus. The graph Laplacian $-\\dv \\grad$ gives rise to the celebrated Laplacian matrix, while the matrix representation of the graph Helmholtzian $\\grad \\grad^* + \\curl^* \\curl$ is called the Helmholtzian matrix. In this paper, we present a new graph-theoretic proof that the Helmholtzian matrix indeed represents the graph Helmholtzian. We then investigate the spectral properties of this matrix. Our main results are as follows: (i) a classification of graphs havin"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present a new graph-theoretic proof that the Helmholtzian matrix indeed represents the graph Helmholtzian. Our main results are as follows: (i) a classification of graphs having exactly two distinct Helmholtzian eigenvalues; (ii) the nullity of the Helmholtzian matrix; and (iii) a combinatorial interpretation of the coefficients of the Helmholtzian polynomial.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The graph-theoretic gradient, curl, and divergence operators are defined so that their adjoints and compositions produce a well-defined Helmholtzian operator whose matrix representation is the one studied.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Helmholtzian matrix on graphs admits a classification of graphs with two eigenvalues, a formula for its nullity, and a combinatorial interpretation of its polynomial coefficients.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A new graph-theoretic proof confirms that the Helmholtzian matrix represents the graph Helmholtzian operator, classifying graphs with exactly two distinct eigenvalues and giving combinatorial meaning to its polynomial coefficients.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5d178054ce9ab4da3650f64fac7d812a8eba49a827a73bc54432171470377bf6"},"source":{"id":"2605.13733","kind":"arxiv","version":1},"verdict":{"id":"a82fe02c-1184-4de9-84b8-a80df8f95313","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:58:45.693332Z","strongest_claim":"We present a new graph-theoretic proof that the Helmholtzian matrix indeed represents the graph Helmholtzian. Our main results are as follows: (i) a classification of graphs having exactly two distinct Helmholtzian eigenvalues; (ii) the nullity of the Helmholtzian matrix; and (iii) a combinatorial interpretation of the coefficients of the Helmholtzian polynomial.","one_line_summary":"The Helmholtzian matrix on graphs admits a classification of graphs with two eigenvalues, a formula for its nullity, and a combinatorial interpretation of its polynomial coefficients.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The graph-theoretic gradient, curl, and divergence operators are defined so that their adjoints and compositions produce a well-defined Helmholtzian operator whose matrix representation is the one studied.","pith_extraction_headline":"A new graph-theoretic proof confirms that the Helmholtzian matrix represents the graph Helmholtzian operator, classifying graphs with exactly two distinct eigenvalues and giving combinatorial meaning to its polynomial coefficients."},"references":{"count":64,"sample":[{"doi":"","year":2017,"title":"K. Adiprasito, J. Huh, E. Katz, Hodge Theory of Matroids, Notices Amer. Math. Soc., 64 (2017), pp. 26–30","work_id":"a1aeb17e-0672-4a2e-aa83-692862ec69fa","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"K. Adiprasito, J. Huh, E. Katz, Hodge theory for combinatorial geometries, Ann. Math., 188 (2018), pp. 381–452","work_id":"da5beba4-b645-49b7-8d21-bf5b5c9ec220","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"L. Bartholdi, T. Schick, N. Smale, S. Smale, Hodge theory on metric spaces, Found. Comput. Math., 12 (2012), pp. 1–48","work_id":"c98132cc-961a-4075-82a8-8f1b4642680a","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"Belardo, Balancedness and the least eigenvalue of Laplacian of signed graphs, Linear Algebra Appl., 446 (2014), pp","work_id":"2db1160d-4485-4236-8182-994e0c38d47b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"F. Belardo, Z. Stani´ c, T. Zaslavsky, Total graph of a signed graph, Ars Math. Contemp., 23 (2023), #P1.02","work_id":"ee9edade-0ba4-4fe6-8f74-a3390cccfccd","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":64,"snapshot_sha256":"cacec8bfdea56f89e35d0f289a6713a0937226f804d397676629b3b639646095","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"28732f33ab7a8632463a83391be988637f3e78981039255126c02ecc966a7ba4"},"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":"2605.13733","created_at":"2026-05-18T02:44:16.539704+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.13733v1","created_at":"2026-05-18T02:44:16.539704+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13733","created_at":"2026-05-18T02:44:16.539704+00:00"},{"alias_kind":"pith_short_12","alias_value":"RTTGLDSE6UPP","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"RTTGLDSE6UPPUZXG","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"RTTGLDSE","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE","json":"https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE.json","graph_json":"https://pith.science/api/pith-number/RTTGLDSE6UPPUZXGP5ZA24MBIE/graph.json","events_json":"https://pith.science/api/pith-number/RTTGLDSE6UPPUZXGP5ZA24MBIE/events.json","paper":"https://pith.science/paper/RTTGLDSE"},"agent_actions":{"view_html":"https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE","download_json":"https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE.json","view_paper":"https://pith.science/paper/RTTGLDSE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.13733&json=true","fetch_graph":"https://pith.science/api/pith-number/RTTGLDSE6UPPUZXGP5ZA24MBIE/graph.json","fetch_events":"https://pith.science/api/pith-number/RTTGLDSE6UPPUZXGP5ZA24MBIE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE/action/storage_attestation","attest_author":"https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE/action/author_attestation","sign_citation":"https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE/action/citation_signature","submit_replication":"https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE/action/replication_record"}},"created_at":"2026-05-18T02:44:16.539704+00:00","updated_at":"2026-05-18T02:44:16.539704+00:00"}