{"paper":{"title":"The stability of independence polynomials of complete bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Guo Chen, Jianhua Tu","submitted_at":"2025-05-30T09:10:28Z","abstract_excerpt":"The independence polynomial of a graph is termed {\\it stable} if all its roots are located in the left half-plane $\\{z \\in \\mathbb{C} : \\mathrm{Re}(z) \\leq 0\\}$, and the graph itself is also referred to as stable. Brown and Cameron (Electron. J. Combin. 25(1) (2018) \\#P1.46) proved that the complete bipartite graph $K_{1,n}$ is stable and posed the question: \\textbf{Are all complete bipartite graphs stable?}\n  We answer this question by establishing the following results:\n  \\begin{itemize}\n  \\item The complete bipartite graphs $K_{2,n}$ and $K_{3,n}$ are stable.\n  \\item For any integer $k\\geq0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.24381","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/2505.24381/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"}