{
  "id": "2210.03236",
  "version": "v1",
  "published": "2022-10-06T22:08:40.000Z",
  "updated": "2022-10-06T22:08:40.000Z",
  "title": "Paley-like graphs over finite fields from vector spaces",
  "authors": [
    "Lucas Reis"
  ],
  "comment": "Accepted for publication in Finite Fields and Their Applications",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Motivated by the well-known Paley graphs over finite fields and their generalizations, in this paper we explore a natural multiplicative-additive analogue of such graphs arising from vector spaces over finite fields. Namely, if $n\\ge 2$ and $U\\subsetneq \\mathbb F_{q^n}$ is an $\\mathbb F_q$-vector space, $G_{U}$ is the (undirected) graph with vertex set $V(G_U)=\\mathbb F_{q^n}$ and edge set $E(G_U)=\\{(a, b)\\in \\mathbb F_{q^n}^2\\,|\\, a\\ne b, ab\\in U\\}$. We describe the structure of an arbitrary maximal clique in $G_U$ and provide bounds on the clique number $\\omega(G_U)$ of $G_U$. In particular, we compute the largest possible value of $\\omega(G_U)$ for arbitrary $q$ and $n$. Moreover, we obtain the exact value of $\\omega(G_U)$ when $U\\subsetneq \\mathbb F_{q^n}$ is any $\\mathbb F_q$-vector space of dimension $d_U\\in \\{1, 2, n-1\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-06T22:08:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "15A63",
      "11T99"
    ],
    "keywords": [
      "vector space",
      "finite fields",
      "paley-like graphs",
      "well-known paley graphs",
      "arbitrary maximal clique"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}