{"paper":{"title":"Sufficient conditions for graphs to be $k$-connected, maximally connected and super-connected","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fuyuan Chen, Lutz Volkmann, Zheng-Jiang Xia, Zhen-Mu Hong","submitted_at":"2017-08-17T18:07:49Z","abstract_excerpt":"Let $G$ be a connected graph with minimum degree $\\delta(G)$ and vertex-connectivity $\\kappa(G)$. The graph $G$ is $k$-connected if $\\kappa(G)\\geq k$, maximally connected if $\\kappa(G) = \\delta(G)$, and super-connected (or super-$\\kappa$) if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is $k$-connected, maximally connected or super-connected if the number of edges or the spectral radius is large enough."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.05396","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":""},"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"}