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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.21866","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-21T01:22:14Z","cross_cats_sorted":["cs.DM","math.NT"],"title_canon_sha256":"628dee5e4d2c1d0aac3e251d29079203034d8091352f8ef77308cbc1b217f1e6","abstract_canon_sha256":"9819557d8ecab651a4425c2cb9509c97cb5df1e2095e6c097e554ba4b79a4e17"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:12.101156Z","signature_b64":"MESaMppB112bmDvOTYNVUkmF/DD5P7OgMK10jkwgMyqCExhP0FFmdUWTEH0T8Er8nJmCW7V+S6FrvtSCn0FgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74740cbdbd2d5ffd60ff1eee535dd0d64fdd582dc8b1e124cd26af8c617303b4","last_reissued_at":"2026-05-22T01:04:12.100479Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:12.100479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $Q_b(X, Y):=X^2+bXY+Y^2, b\\ne 0$. 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