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Motivated by this, we define the Hermitian-Randi\\'c matrix $R_{H}(M)=(r_{h})_{kl}$ of a mixed graph $M$, where $(r_{h})_{kl}=-(r_{h})_{lk}=\\frac{\\textbf{i}}{\\sqrt{d_{k}d_{l}}}$ ($\\textbf{i}=\\sqrt{-1}$) if $(v_{k},v_{l})$ is an arc of $M$, $(r_{h})_{kl}=(r_{h})_{lk}=\\frac{1}{\\sqrt{d_{k}d_{l}}}$ if $v_{k}v_{l}$ is an undirected edge of $M$, and $(r_{h})_{kl}=0$ otherwise"},"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":"1610.09783","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-31T04:13:31Z","cross_cats_sorted":[],"title_canon_sha256":"b088502aaa30d1a7a4d49e03b823b87212dcd9080fcd558fa8656f3260c43071","abstract_canon_sha256":"4177f99a8c8c57be52cab63e0dfb1ff64497358b93bc22215d9fc9b648f3f0f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:47.194543Z","signature_b64":"aM/4+vryYNQhyPlYA+z/YYRHkClF5RH4P5wnp/m4T2CIssDLqseUi1qLkrvgaHBA+7+SmBvfWGto6H5kny5YDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7fb14e7e7e5e3b8da5d471a55321776320df576e29b9245f39dc492789a01aaa","last_reissued_at":"2026-05-18T01:00:47.194108Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:47.194108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hermitian-Randi\\'c matrix and Hermitian-Randi\\'c energy of mixed graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ligong Wang, Qiannan Zhou, Yong Lu","submitted_at":"2016-10-31T04:13:31Z","abstract_excerpt":"Let $M$ be a mixed graph and $H(M)$ be its Hermitian-adjacency matrix. 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Motivated by this, we define the Hermitian-Randi\\'c matrix $R_{H}(M)=(r_{h})_{kl}$ of a mixed graph $M$, where $(r_{h})_{kl}=-(r_{h})_{lk}=\\frac{\\textbf{i}}{\\sqrt{d_{k}d_{l}}}$ ($\\textbf{i}=\\sqrt{-1}$) if $(v_{k},v_{l})$ is an arc of $M$, $(r_{h})_{kl}=(r_{h})_{lk}=\\frac{1}{\\sqrt{d_{k}d_{l}}}$ if $v_{k}v_{l}$ is an undirected edge of $M$, and $(r_{h})_{kl}=0$ otherwise"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.09783","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":"1610.09783","created_at":"2026-05-18T01:00:47.194169+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.09783v1","created_at":"2026-05-18T01:00:47.194169+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.09783","created_at":"2026-05-18T01:00:47.194169+00:00"},{"alias_kind":"pith_short_12","alias_value":"P6YU47T6LY5Y","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_16","alias_value":"P6YU47T6LY5Y3JOU","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_8","alias_value":"P6YU47T6","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/P6YU47T6LY5Y3JOUOGSVGILXMM","json":"https://pith.science/pith/P6YU47T6LY5Y3JOUOGSVGILXMM.json","graph_json":"https://pith.science/api/pith-number/P6YU47T6LY5Y3JOUOGSVGILXMM/graph.json","events_json":"https://pith.science/api/pith-number/P6YU47T6LY5Y3JOUOGSVGILXMM/events.json","paper":"https://pith.science/paper/P6YU47T6"},"agent_actions":{"view_html":"https://pith.science/pith/P6YU47T6LY5Y3JOUOGSVGILXMM","download_json":"https://pith.science/pith/P6YU47T6LY5Y3JOUOGSVGILXMM.json","view_paper":"https://pith.science/paper/P6YU47T6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.09783&json=true","fetch_graph":"https://pith.science/api/pith-number/P6YU47T6LY5Y3JOUOGSVGILXMM/graph.json","fetch_events":"https://pith.science/api/pith-number/P6YU47T6LY5Y3JOUOGSVGILXMM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P6YU47T6LY5Y3JOUOGSVGILXMM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P6YU47T6LY5Y3JOUOGSVGILXMM/action/storage_attestation","attest_author":"https://pith.science/pith/P6YU47T6LY5Y3JOUOGSVGILXMM/action/author_attestation","sign_citation":"https://pith.science/pith/P6YU47T6LY5Y3JOUOGSVGILXMM/action/citation_signature","submit_replication":"https://pith.science/pith/P6YU47T6LY5Y3JOUOGSVGILXMM/action/replication_record"}},"created_at":"2026-05-18T01:00:47.194169+00:00","updated_at":"2026-05-18T01:00:47.194169+00:00"}