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In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $\\rho(G)\\cup (cd(G)\\setminus\\{1\\})$, such that an element $p$ of $\\rho(G)$ is adjacent to an element $m$ of $cd(G)\\setminus\\{1\\}$ if and only if $p$ divides $m$. We denote this graph simply by $B(G)$. Then by means of combinato"},"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":"1511.07644","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-11-24T11:02:23Z","cross_cats_sorted":[],"title_canon_sha256":"63767ebb3f8c6464ff03735b92bc14ad14aa04bf04abe34582691fb1ce8e967d","abstract_canon_sha256":"8385089b3e0a350eaa8eef8094914ab6ae8f1621d62fac82c3fa3d96611368e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:04.090140Z","signature_b64":"MudVtaMuRkVOJuIkSyFa7Lm3ZcEUWAUnq8J29tLoBDTwbSGN7OoG4hgf9PMrGuQA2oBd5J05GJtRaLlO2dWbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58adf09aad9d9b1c8f9d38443232a39f44a7fd4964187fb3fa734451a036acb4","last_reissued_at":"2026-05-18T01:26:04.089551Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:04.089551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bipartite divisor graph for the set of irreducible character degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Roghayeh Hafezieh","submitted_at":"2015-11-24T11:02:23Z","abstract_excerpt":"Let $G$ be a finite group. We consider the set of the irreducible complex characters of $G$, namely $Irr(G)$, and the related degree set $cd(G)=\\{\\chi(1) : \\chi\\in Irr(G)\\}$. Let $\\rho(G)$ be the set of all primes which divide some character degree of $G$. In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $\\rho(G)\\cup (cd(G)\\setminus\\{1\\})$, such that an element $p$ of $\\rho(G)$ is adjacent to an element $m$ of $cd(G)\\setminus\\{1\\}$ if and only if $p$ divides $m$. We denote this graph simply by $B(G)$. Then by means of combinato"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07644","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":"1511.07644","created_at":"2026-05-18T01:26:04.089641+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.07644v1","created_at":"2026-05-18T01:26:04.089641+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.07644","created_at":"2026-05-18T01:26:04.089641+00:00"},{"alias_kind":"pith_short_12","alias_value":"LCW7BGVNTWNR","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_16","alias_value":"LCW7BGVNTWNRZD45","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_8","alias_value":"LCW7BGVN","created_at":"2026-05-18T12:29:29.992203+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/LCW7BGVNTWNRZD45HBCDEMVDT5","json":"https://pith.science/pith/LCW7BGVNTWNRZD45HBCDEMVDT5.json","graph_json":"https://pith.science/api/pith-number/LCW7BGVNTWNRZD45HBCDEMVDT5/graph.json","events_json":"https://pith.science/api/pith-number/LCW7BGVNTWNRZD45HBCDEMVDT5/events.json","paper":"https://pith.science/paper/LCW7BGVN"},"agent_actions":{"view_html":"https://pith.science/pith/LCW7BGVNTWNRZD45HBCDEMVDT5","download_json":"https://pith.science/pith/LCW7BGVNTWNRZD45HBCDEMVDT5.json","view_paper":"https://pith.science/paper/LCW7BGVN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.07644&json=true","fetch_graph":"https://pith.science/api/pith-number/LCW7BGVNTWNRZD45HBCDEMVDT5/graph.json","fetch_events":"https://pith.science/api/pith-number/LCW7BGVNTWNRZD45HBCDEMVDT5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LCW7BGVNTWNRZD45HBCDEMVDT5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LCW7BGVNTWNRZD45HBCDEMVDT5/action/storage_attestation","attest_author":"https://pith.science/pith/LCW7BGVNTWNRZD45HBCDEMVDT5/action/author_attestation","sign_citation":"https://pith.science/pith/LCW7BGVNTWNRZD45HBCDEMVDT5/action/citation_signature","submit_replication":"https://pith.science/pith/LCW7BGVNTWNRZD45HBCDEMVDT5/action/replication_record"}},"created_at":"2026-05-18T01:26:04.089641+00:00","updated_at":"2026-05-18T01:26:04.089641+00:00"}