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The {\\it $\\alpha_{v}$-cover} number of a graph, denoted by $\\alpha_{v}(G)$, is the maximum natural number $m$ such that every vertex of $G$ belongs to a maximal independent set with at least $m$ vertices. In the first part of this paper we prove that $\\alpha(G)\\leq \\tau(G)[1+\\alpha(G)-\\alpha_{v}(G)]$. We also discuss some conjectures analogous to this theorem.\n  In the second part we give a lower bound for the number of edges of a graph $G$ as a function of the stability "},"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":"math/0510387","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2005-10-18T17:44:23Z","cross_cats_sorted":[],"title_canon_sha256":"17c3bd45e37d6f48315bbf78d60c588b54591d52256ca9877e86921d2567424f","abstract_canon_sha256":"9f8c5ca8b964612d9b828066ff7e79a2759d984cde139195afceda92b6ec4bf7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:14:30.744835Z","signature_b64":"iqGVcFTySfO9qeLjkV4b0MKkl0ymicIxYwjZ3CUftuytOQZiCgf+mP4wPeBEfHYQoSwOb5WneG5EsxUiwy9OAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5c04d43ab4f3418dcc76de654129c31caadd34dd1f646176857b40a03133946","last_reissued_at":"2026-05-18T03:14:30.744299Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:14:30.744299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On bounds for some graph invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carlos E. 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