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Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in $\\ell_n^m$ with $n=2m$. In this paper, we characterize the case $n\\geq 2m+1$. This confirms the conjecture of Li and Zhang(S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012, 4400--4411). Further, we characterize the unique grap"},"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":"1511.06902","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2015-11-21T17:15:01Z","cross_cats_sorted":[],"title_canon_sha256":"a40fadc3c34f027b8dbba2bce78db405dda5974273333b8d8663555fc1397481","abstract_canon_sha256":"b6235cf278fd01f6f1b08737fed82cac391316f171686e2afc73a5e5d10b2eda"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:17.199811Z","signature_b64":"7EGQI+lm4sFxIqIbASiVnrN4ajH0AWXU1bk14J6OJL+jgsJXC+lnTjsD3oxjrAvfhuhy0qeBwRbEdprggv+CAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"324b1c319a17ab8c5970db3f10466dd0511e37882df84fcdd55ca17e6ad5afee","last_reissued_at":"2026-05-18T01:26:17.199239Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:17.199239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a conjecture for the signless Laplacian spectral radius of cacti with given matching number","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lihua You, Minjie Zhang, Shuchao Li, Yun Shen","submitted_at":"2015-11-21T17:15:01Z","abstract_excerpt":"A connected graph $G$ is a cactus if any two of its cycles have at most one common vertex. Let $\\ell_n^m$ be the set of cacti on $n$ vertices with matching number $m.$ S.C. Li and M.J. Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in $\\ell_n^m$ with $n=2m$. In this paper, we characterize the case $n\\geq 2m+1$. This confirms the conjecture of Li and Zhang(S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012, 4400--4411). 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