{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:PQUHL6E6PLMHK34RMYXBQ2USBP","short_pith_number":"pith:PQUHL6E6","schema_version":"1.0","canonical_sha256":"7c2875f89e7ad8756f91662e186a920bf87b27c8fd6319bf5f8b2be65648231e","source":{"kind":"arxiv","id":"1609.01425","version":1},"attestation_state":"computed","paper":{"title":"Vertex weighted Laplacian graph energy and other topological indices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"H. Panahbar, Reza Sharafdini","submitted_at":"2016-09-06T08:11:56Z","abstract_excerpt":"Let $G$ be a graph with a vertex weight $\\omega$ and the vertices $v_1,\\ldots,v_n$. The Laplacian matrix of $G$ with respect to $\\omega$ is defined as\n  $L_\\omega(G)=\\mathrm{diag}(\\omega(v_1),\\cdots,\\omega(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $\\mu_1,\\cdots,\\mu_n$ be eigenvalues of $L_\\omega(G)$. Then the Laplacian energy of $G$ with respect to $\\omega$ defined as $LE_\\omega (G)=\\sum_{i=1}^n\\big|\\mu_i - \\bar{\\omega}\\big|$, where $\\bar{\\omega}$ is the average of $\\omega$, i.e., $\\bar{\\omega}=\\dfrac{\\sum_{i=1}^{n}\\omega(v_i)}{n}$. In this paper we consider several natural"},"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":"1609.01425","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-06T08:11:56Z","cross_cats_sorted":[],"title_canon_sha256":"d9e3cb1f8670ff6bfc39d3da28c7608e3af3c8fcc2c4b8f3d3f4885775fd2feb","abstract_canon_sha256":"4753cdd6efeffeaa49e9a84664af3c3e32715da43549eaf7aaf450111c24f1b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:42.901133Z","signature_b64":"KmylzOFMk5nAMmfb8I+W3W5iSXLSz0I/RH1SvrsvtenZiagnUA11qE0EpJGn1SiA8J8O3b1rYwqp1GAIfn2WBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c2875f89e7ad8756f91662e186a920bf87b27c8fd6319bf5f8b2be65648231e","last_reissued_at":"2026-05-18T01:04:42.900519Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:42.900519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vertex weighted Laplacian graph energy and other topological indices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"H. Panahbar, Reza Sharafdini","submitted_at":"2016-09-06T08:11:56Z","abstract_excerpt":"Let $G$ be a graph with a vertex weight $\\omega$ and the vertices $v_1,\\ldots,v_n$. The Laplacian matrix of $G$ with respect to $\\omega$ is defined as\n  $L_\\omega(G)=\\mathrm{diag}(\\omega(v_1),\\cdots,\\omega(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $\\mu_1,\\cdots,\\mu_n$ be eigenvalues of $L_\\omega(G)$. Then the Laplacian energy of $G$ with respect to $\\omega$ defined as $LE_\\omega (G)=\\sum_{i=1}^n\\big|\\mu_i - \\bar{\\omega}\\big|$, where $\\bar{\\omega}$ is the average of $\\omega$, i.e., $\\bar{\\omega}=\\dfrac{\\sum_{i=1}^{n}\\omega(v_i)}{n}$. In this paper we consider several natural"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01425","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":""},"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":"1609.01425","created_at":"2026-05-18T01:04:42.900590+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.01425v1","created_at":"2026-05-18T01:04:42.900590+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.01425","created_at":"2026-05-18T01:04:42.900590+00:00"},{"alias_kind":"pith_short_12","alias_value":"PQUHL6E6PLMH","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_16","alias_value":"PQUHL6E6PLMHK34R","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_8","alias_value":"PQUHL6E6","created_at":"2026-05-18T12:30:39.010887+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/PQUHL6E6PLMHK34RMYXBQ2USBP","json":"https://pith.science/pith/PQUHL6E6PLMHK34RMYXBQ2USBP.json","graph_json":"https://pith.science/api/pith-number/PQUHL6E6PLMHK34RMYXBQ2USBP/graph.json","events_json":"https://pith.science/api/pith-number/PQUHL6E6PLMHK34RMYXBQ2USBP/events.json","paper":"https://pith.science/paper/PQUHL6E6"},"agent_actions":{"view_html":"https://pith.science/pith/PQUHL6E6PLMHK34RMYXBQ2USBP","download_json":"https://pith.science/pith/PQUHL6E6PLMHK34RMYXBQ2USBP.json","view_paper":"https://pith.science/paper/PQUHL6E6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.01425&json=true","fetch_graph":"https://pith.science/api/pith-number/PQUHL6E6PLMHK34RMYXBQ2USBP/graph.json","fetch_events":"https://pith.science/api/pith-number/PQUHL6E6PLMHK34RMYXBQ2USBP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PQUHL6E6PLMHK34RMYXBQ2USBP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PQUHL6E6PLMHK34RMYXBQ2USBP/action/storage_attestation","attest_author":"https://pith.science/pith/PQUHL6E6PLMHK34RMYXBQ2USBP/action/author_attestation","sign_citation":"https://pith.science/pith/PQUHL6E6PLMHK34RMYXBQ2USBP/action/citation_signature","submit_replication":"https://pith.science/pith/PQUHL6E6PLMHK34RMYXBQ2USBP/action/replication_record"}},"created_at":"2026-05-18T01:04:42.900590+00:00","updated_at":"2026-05-18T01:04:42.900590+00:00"}