{"paper":{"title":"Proof of a conjecture on monomial graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Felix Lazebnik, Stephen D. Lappano, Xiang-dong Hou","submitted_at":"2015-07-19T16:50:14Z","abstract_excerpt":"Let $e$ be a positive integer, $p$ be an odd prime, $q=p^{e}$, and $\\Bbb F_q$ be the finite field of $q$ elements. Let $f,g \\in \\Bbb F_q [X,Y]$. The graph $G=G_q(f,g)$ is a bipartite graph with vertex partitions $P=\\Bbb F_q^3$ and $L=\\Bbb F_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\\in P$ is adjacent to a vertex $[l] = [l_1,l_2,l_3]\\in L$ if and only if $p_2 + l_2 = f(p_1,l_1)$ and $p_3 + l_3 = g(p_1,l_1)$. Motivated by some questions in finite geometry and extremal graph theory, Dmytrenko, Lazebnik and Williford conjectured in 2007 that if $f$ and $g$ are both monomials a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05306","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"}