{"paper":{"title":"Line graphs with the largest eigenvalue multiplicity","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Dein Wong, Songnian Xu, Wenhao Zhen","submitted_at":"2024-11-22T10:17:33Z","abstract_excerpt":"For a connected graph $G$, we denote by $L(G)$, $m_{G}(\\lambda)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $\\lambda$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In 2023, Yang et al. \\cite{WL LT} proved that $m_{L(T)}(\\lambda)\\leq p(T)-1$ for any tree $T$ with $p(T)\\geq 3$, and characterized all trees $T$ with $m_{L(T)}(\\lambda) = p(T)-1$. In 2024, Chang et al. \\cite{-1 LG} proved that, if $G$ is not a cycle, then $m_{L(G)}(\\lambda)\\leq 2c(G)+p(G)-1$, and characterized all graphs $G$ with $m_{L(G)}(-1) = 2c(G)+p(G)-1$. Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14835","kind":"arxiv","version":2},"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/2411.14835/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"}