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We show that $\\Gamma_s (G)$ is not a star graph, a tree, an $n$-partite graph for any positive integer $n \\geq 2$ and not a regular graph for any non-solvable finite group $G$. We compute the girth of $\\Gamma_s (G)$ and derive a lower bound of the clique number of $\\Gamma_s (G)$. We prove the non-existence of finite non-solvable groups wh"},"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":"1903.01755","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-03-05T10:14:16Z","cross_cats_sorted":[],"title_canon_sha256":"017d048ab9e0eeafdf88310fa37334e0f464f2166cb192ac3e5b468d3d84861c","abstract_canon_sha256":"fa55f437e23b375d4a9aa17bc6be01a8e9778559c3dc16f5df408950ebea6791"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:59.854033Z","signature_b64":"6KE0t8NAze4QM9dMfIRYUZfE6CDz4PGTR98fQ7t+T0hlgtIGGlq+Vpp6GLD7S4V2aMBgNUtQFxcIRLbhmo9cCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a256b4536ad837c41f1366a6cd8410d11131df4fbeef7fc455406c38bad62ab","last_reissued_at":"2026-05-17T23:51:59.853629Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:59.853629Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on solvable graphs of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Deiborlang Nongsiang, Parthajit Bhowal, Rajat Kanti Nath","submitted_at":"2019-03-05T10:14:16Z","abstract_excerpt":"Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph $\\Gamma_s(G)$ of $G$ is a graph with vertex set $G\\setminus Sol(G)$ and two distinct vertices $u$ and $v$ are adjacent if and only if $\\langle u, v \\rangle$ is solvable. We show that $\\Gamma_s (G)$ is not a star graph, a tree, an $n$-partite graph for any positive integer $n \\geq 2$ and not a regular graph for any non-solvable finite group $G$. We compute the girth of $\\Gamma_s (G)$ and derive a lower bound of the clique number of $\\Gamma_s (G)$. We prove the non-existence of finite non-solvable groups wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.01755","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":"1903.01755","created_at":"2026-05-17T23:51:59.853696+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.01755v1","created_at":"2026-05-17T23:51:59.853696+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.01755","created_at":"2026-05-17T23:51:59.853696+00:00"},{"alias_kind":"pith_short_12","alias_value":"PISWWRJWVWBX","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_16","alias_value":"PISWWRJWVWBXYQPR","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_8","alias_value":"PISWWRJW","created_at":"2026-05-18T12:33:24.271573+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/PISWWRJWVWBXYQPRGZVGZWCBBU","json":"https://pith.science/pith/PISWWRJWVWBXYQPRGZVGZWCBBU.json","graph_json":"https://pith.science/api/pith-number/PISWWRJWVWBXYQPRGZVGZWCBBU/graph.json","events_json":"https://pith.science/api/pith-number/PISWWRJWVWBXYQPRGZVGZWCBBU/events.json","paper":"https://pith.science/paper/PISWWRJW"},"agent_actions":{"view_html":"https://pith.science/pith/PISWWRJWVWBXYQPRGZVGZWCBBU","download_json":"https://pith.science/pith/PISWWRJWVWBXYQPRGZVGZWCBBU.json","view_paper":"https://pith.science/paper/PISWWRJW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.01755&json=true","fetch_graph":"https://pith.science/api/pith-number/PISWWRJWVWBXYQPRGZVGZWCBBU/graph.json","fetch_events":"https://pith.science/api/pith-number/PISWWRJWVWBXYQPRGZVGZWCBBU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PISWWRJWVWBXYQPRGZVGZWCBBU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PISWWRJWVWBXYQPRGZVGZWCBBU/action/storage_attestation","attest_author":"https://pith.science/pith/PISWWRJWVWBXYQPRGZVGZWCBBU/action/author_attestation","sign_citation":"https://pith.science/pith/PISWWRJWVWBXYQPRGZVGZWCBBU/action/citation_signature","submit_replication":"https://pith.science/pith/PISWWRJWVWBXYQPRGZVGZWCBBU/action/replication_record"}},"created_at":"2026-05-17T23:51:59.853696+00:00","updated_at":"2026-05-17T23:51:59.853696+00:00"}