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The independence polynomial has the notable feature of being essentially closed under graph composition (lexicographic product). In this paper, we determine the independence polynomials of size five. For a disconnected graph $G$, we exploit the fact that $I_G(z)$ factors as the product of the independence polynomials of the connected components of $G$"},"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":"2607.29322","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-31T11:53:38Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"19c28bf40ec72035cc5653f0fe4ce5bef6f281c890b9d25d6a7d0e05b5706aba","abstract_canon_sha256":"4a85cc18d9c42c8855733ac80e918377543c0a85b4fe27dfb223ced8a756e608"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-03T01:24:53.104088Z","signature_b64":"CsEYCNl1g2Z02GnJprOt8fVEM+9Aq6iSHRlGorFUmYI8rh9dkOk7Qcfectjw8JQ2GGcpnixe01Nyf14AVcnqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b54780ebeb841d2bfb96fa995b1a86aa19365339af5c650f8cd88f813493ebb","last_reissued_at":"2026-08-03T01:24:53.102654Z","signature_status":"signed_v1","first_computed_at":"2026-08-03T01:24:53.102654Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Structural Classification of a Graph with Independence Number Five","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.CO","authors_text":"Dinesh Kumar, Manisha Kumari","submitted_at":"2026-07-31T11:53:38Z","abstract_excerpt":"The independence polynomial of a simple graph $G$ is given by \\( I_G(z) = i_0 + i_1 z + i_2 z^2 + \\cdots + i_\\alpha z^\\alpha \\), where \\( i_\\alpha \\) denotes the size of a maximum independent set, also called the independence number of the graph. The independence polynomial has the notable feature of being essentially closed under graph composition (lexicographic product). In this paper, we determine the independence polynomials of size five. 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