{"paper":{"title":"The multiplicity of the laplacian eigenvalue 1 of a tree","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Dein Wong, Songnian Xu, Wenhao Zhen","submitted_at":"2026-05-29T08:18:32Z","abstract_excerpt":"Let $G$ be a connected, undirected simple graph. Denote by $L(G)$ the Laplacian matrix of $G$, and let $m_{G}(\\lambda)$ be the multiplicity of an eigenvalue $\\lambda$ of $L(G)$. When $G$ is a tree $T$ with $n \\ge 6$ vertices, Tian et al. [Discrete Mathematics, 2026] proved that if $T$ is reduced and contains no pendant $P_3$, then \\[ m_{T}(1) \\le \\frac{n-6}{4}, \\] and they gave a complete characterization of the graphs for which equality holds.\n  In this paper, we further investigate the above problem. Still assuming that $T$ is a tree with $n \\ge 7$ vertices which is reduced and has no pendan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30982","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.30982/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}