{"paper":{"title":"Spectral skeletons and applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wenqian Zhang","submitted_at":"2025-01-24T03:50:05Z","abstract_excerpt":"For a graph $G$, its spectral radius $\\rho(G)$ is the largest eigenvalue of its adjacency matrix. Let $\\mathcal{F}$ be a finite family of graphs with $\\min_{F\\in \\mathcal{F}}\\chi(F)=r+1\\geq3$, where $\\chi(F)$ is the chromatic number of $F$. Set $t=\\max_{F\\in\\mathcal{F}}|F|$. Let $T(rt,r)$ be the Tur\\'{a}n graph of order $rt$ with $r$ parts. Assume that some $F_{0}\\subseteq\\mathcal{F}$ is a subgraph of the graph obtained from $T(rt,r)$ by embedding a path or a matching in one part. Let ${\\rm EX}(n,\\mathcal{F})$ be the set of graphs with the maximum number of edges among all the graphs of order "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.14218","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/2501.14218/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"}