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In this paper we introduce $\\widetilde{SL}(G)=\\widetilde{D}(G)-S(G)$ to be a new kind of skew Laplacian matrix of $G$, where $\\widetilde{D}(G)=D^+(G)-D^-(G)$ and $S(G)$ is the skew-adjacency matrix of $G$, and from which we define the skew Laplacian energy $SLE(G)$ of $G$ as the sum of the norms of all the eigenvalues of $\\widet"},"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":"1304.6465","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-24T02:18:21Z","cross_cats_sorted":[],"title_canon_sha256":"490438a3fe916674853e9575da9c2002827f15e78ee9e50899f35504fe3174d9","abstract_canon_sha256":"0c799eb003b3ab9cdbbb71dcd8d16299010e9836c47346ff416b5cc1a45b37e8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:13.044643Z","signature_b64":"oUxgb1ZUofw3PPFFb3UlaycBcyJweucDG4aLmZ57yNkyEjwWTs2B3o96koQNH9XOQdzgvwXR/7ClhE1pPPaeCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a804fe225a2051b2b8fc10f8099502aa1122bac2d1fc68e82d76f023bae6f08d","last_reissued_at":"2026-05-18T03:27:13.043937Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:13.043937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New skew Laplacian energy of a simple digraph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiangli Song, Qingqiong Cai, Xueliang Li","submitted_at":"2013-04-24T02:18:21Z","abstract_excerpt":"For a simple digraph $G$ of order $n$ with vertex set $\\{v_1,v_2,\\ldots, v_n\\}$, let $d_i^+$ and $d_i^-$ denote the out-degree and in-degree of a vertex $v_i$ in $G$, respectively. Let $D^+(G)=diag(d_1^+,d_2^+,\\ldots,d_n^+)$ and $D^-(G)=diag(d_1^-,d_2^-,\\ldots,d_n^-)$. In this paper we introduce $\\widetilde{SL}(G)=\\widetilde{D}(G)-S(G)$ to be a new kind of skew Laplacian matrix of $G$, where $\\widetilde{D}(G)=D^+(G)-D^-(G)$ and $S(G)$ is the skew-adjacency matrix of $G$, and from which we define the skew Laplacian energy $SLE(G)$ of $G$ as the sum of the norms of all the eigenvalues of $\\widet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.6465","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":"1304.6465","created_at":"2026-05-18T03:27:13.044044+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.6465v1","created_at":"2026-05-18T03:27:13.044044+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.6465","created_at":"2026-05-18T03:27:13.044044+00:00"},{"alias_kind":"pith_short_12","alias_value":"VACP4IS2EBI3","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"VACP4IS2EBI3FOH4","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"VACP4IS2","created_at":"2026-05-18T12:28:04.890932+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.23708","citing_title":"Learning Dynamic Stability Landscapes in Synchronization Networks","ref_index":103,"is_internal_anchor":true},{"citing_arxiv_id":"2604.18017","citing_title":"A Unified Theory of Edge Weights: Stability of General Laplacian Networks from Matrix Phases and Asymmetry Rayleigh Ratios","ref_index":26,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VACP4IS2EBI3FOH4CD4ATFICVI","json":"https://pith.science/pith/VACP4IS2EBI3FOH4CD4ATFICVI.json","graph_json":"https://pith.science/api/pith-number/VACP4IS2EBI3FOH4CD4ATFICVI/graph.json","events_json":"https://pith.science/api/pith-number/VACP4IS2EBI3FOH4CD4ATFICVI/events.json","paper":"https://pith.science/paper/VACP4IS2"},"agent_actions":{"view_html":"https://pith.science/pith/VACP4IS2EBI3FOH4CD4ATFICVI","download_json":"https://pith.science/pith/VACP4IS2EBI3FOH4CD4ATFICVI.json","view_paper":"https://pith.science/paper/VACP4IS2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.6465&json=true","fetch_graph":"https://pith.science/api/pith-number/VACP4IS2EBI3FOH4CD4ATFICVI/graph.json","fetch_events":"https://pith.science/api/pith-number/VACP4IS2EBI3FOH4CD4ATFICVI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VACP4IS2EBI3FOH4CD4ATFICVI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VACP4IS2EBI3FOH4CD4ATFICVI/action/storage_attestation","attest_author":"https://pith.science/pith/VACP4IS2EBI3FOH4CD4ATFICVI/action/author_attestation","sign_citation":"https://pith.science/pith/VACP4IS2EBI3FOH4CD4ATFICVI/action/citation_signature","submit_replication":"https://pith.science/pith/VACP4IS2EBI3FOH4CD4ATFICVI/action/replication_record"}},"created_at":"2026-05-18T03:27:13.044044+00:00","updated_at":"2026-05-18T03:27:13.044044+00:00"}