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Let $S_{n,k}$ be the graph obtained by joining each vertex of a complete graph of order $k$ to each vertex of an independent set of order $n-k;$ and let $S_{n,k}^{+}$ be the graph obtained by adding an edge to $S_{n,k}.$\n  It is shown that if $k\\geq2,$ $n\\geq400k^{2},$ and $G$ is a graph of order $n,$ with no cycle of length $2k+2,$ then $q\\left( G\\right) <q\\left( S_{n,k}^{+}\\right) ,$ unless $G=S_{n,k}^{+}.$ This result completes the proof of a conjecture of de Freitas, Niki"},"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":"1410.2142","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-05T10:01:26Z","cross_cats_sorted":[],"title_canon_sha256":"6787815035ddbda514c646e566e69e47e5b168f2e7b6c6b7ed569b2b3eea8dca","abstract_canon_sha256":"bbc4364c11910e621edf1833df7f812cdc81cd4c7cce76271ab76239c71e4918"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:48.118369Z","signature_b64":"kEdkgHqcL0Z5lIYSubfLC8uF0I7GBtjcXKNi3G/9eRvRLx/tm58sr/ttJgPl7gdtgcz29BYPQQuxsLDNTmVZCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"946dc927db9498aacfc28cc24f581a3e70d26bc0ae5bf25753098d62e42108e5","last_reissued_at":"2026-05-18T02:40:48.117743Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:48.117743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maxima of the Q-index: forbidden even cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vladimir Nikiforov, Xiying Yuan","submitted_at":"2014-10-05T10:01:26Z","abstract_excerpt":"Let $G$ be a graph of order $n$ and let $q\\left( G\\right) $ be the largest eigenvalue of the signless Laplacian of $G$. 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