{"paper":{"title":"Amply regular graphs with $\\mu$ close to half the valency and group divisible designs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chenhui Lv, Jack H. Koolen, Wei Jin","submitted_at":"2026-05-25T12:06:38Z","abstract_excerpt":"In this paper, we classify connected amply regular graphs with diameter $d \\geq 4$ and parameters $(v, k, \\lambda, \\mu)$ satisfying $\\mu = \\frac{k-1}{2}$, where $k\\geq 5$ is odd. We prove that such a graph must be exactly one of the following: the $5$-cube, the graph $\\K_2 \\square \\Lambda$, where $\\Lambda$ is the unique bipartite $(0,2)$-graph on $14$ vertices, or the point--block incidence graph of a group divisible design with the dual property, namely a $GDDDP\\left(2, k+1;\\, k;\\, 0, \\frac{k-1}{2}\\right)$. For the last family, we give equivalent characterizations in terms of bipartite $Q$-re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25754","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/2605.25754/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"}