{"paper":{"title":"Graphs from quadratic forms and vector spaces over finite fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","math.NT"],"primary_cat":"math.CO","authors_text":"Jean Godard, Lucas Reis","submitted_at":"2026-05-21T01:22:14Z","abstract_excerpt":"Let $q$ be an odd prime power, let $n\\ge 2$, and let $V\\subsetneq \\mathbb F_{q^n}$ be a proper $\\mathbb F_q$-vector subspace. Given a nonzero quadratic form $Q(X,Y)\\in \\mathbb F_{q^n}[X,Y]$, we consider the graph $\\Gamma(Q,V)$ that naturally arises from the condition $Q(X,Y)\\in V$. We determine all quadratic forms $Q$ for which $\\Gamma(Q,V)$ is undirected for every $V$. Besides the case $Q(x,y)=XY$, studied earlier by the second author, this essentially leads to the forms $X^2\\pm Y^2$ and the family $Q_b(X, Y):=X^2+bXY+Y^2, b\\ne 0$. We then study connectedness and clique number for the corresp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21866","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/2605.21866/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"}