{"paper":{"title":"Nonregular graphs with a given maximum degree attaining maximum spectral radius","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chenxi Yang, Jiahui Liu, Zejun Huang","submitted_at":"2024-11-26T12:28:51Z","abstract_excerpt":"Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $\\Delta$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(\\Delta,\\ldots,\\Delta,\\delta)$ with $\\delta<\\Delta$. For $\\Delta=3$ and $\\Delta=4$, Liu (2024) confirmed this conjecture by characterizing the structure of such graphs. Liu also proposed a modified version of the conjecture for fixed $\\Delta$ and sufficiently large $n$, stating that the above $\\delta=\\Delta-1$ if $\\Delta$ and $n$ are both odd, $\\delta=1$ if $\\Delta$ is odd and $n$ is even, an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.17371","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/2411.17371/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"}