{"paper":{"title":"Open problem on $\\sigma$-invariant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kinkar Ch. Das, Seyed Ahmad Mojallal","submitted_at":"2017-11-18T18:08:06Z","abstract_excerpt":"Let $G$ be a graph of order $n$ with $m$ edges. Also let $\\mu_1\\geq \\mu_2\\geq \\cdots\\geq \\mu_{n-1}\\geq \\mu_n=0$ be the Laplacian eigenvalues of graph $G$ and let $\\sigma=\\sigma(G)$ $(1\\leq \\sigma\\leq n)$ be the largest positive integer such that $\\mu_{\\sigma}\\geq \\frac{2m}{n}$. In this paper, we prove that $\\mu_2(G)\\geq \\frac{2m}{n}$ for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in \\cite{KMT}, that is, the characterization of all graphs with $\\sigma=1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.06906","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":""},"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"}