{"paper":{"title":"Spectral and Combinatorial Properties of Some Algebraically Defined Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Felix Lazebnik, Sebastian M. Cioab\\u{a}, Shuying Sun","submitted_at":"2017-08-25T01:52:31Z","abstract_excerpt":"Let $k\\ge 3$ be an integer, $q$ be a prime power, and $\\mathbb{F}_q$ denote the field of $q$ elements. Let $f_i, g_i\\in\\mathbb{F}_q[X]$, $3\\le i\\le k$, such that $g_i(-X) = -\\, g_i(X)$. We define a graph $S(k,q) = S(k,q;f_3,g_3,\\cdots,f_k,g_k)$ as a graph with the vertex set $\\mathbb{F}_q^k$ and edges defined as follows: vertices $a = (a_1,a_2,\\ldots,a_k)$ and $b = (b_1,b_2,\\ldots,b_k)$ are adjacent if $a_1\\ne b_1$ and the following $k-2$ relations on their components hold: $$ b_i-a_i = g_i(b_1-a_1)f_i\\Bigl(\\frac{b_2-a_2}{b_1-a_1}\\Bigr)\\;,\\quad 3\\le i\\le k. $$ We show that graphs $S(k,q)$ gene"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.07597","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"}