{"paper":{"title":"The Merrifield-Simmons conjecture also holds for parity graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Martin Trinks","submitted_at":"2014-01-08T08:11:33Z","abstract_excerpt":"The Merrifield-Simmons conjectures states a relation between the distance of vertices in a simple graph $G$ and the number of independent sets, denoted as $\\sigma(G)$, in vertex-deleted subgraphs. Namely, that the sign of the term $\\sigma(G_{-u}) \\cdot \\sigma(G_{-v}) - \\sigma(G) \\cdot \\sigma(G_{-u-v})$ only depends on the parity of the distance of $u$ and $v$ in $G$. We prove this statement in the case of parity graphs and give some evidence that this result may not be further generalized to other classes of graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1596","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"}