Paley-like graphs over finite fields from vector spaces

TL;DR

This paper introduces a new class of graphs over finite fields based on vector space structures, analyzes their clique properties, and determines maximum clique sizes for specific subspace dimensions.

Contribution

It defines a multiplicative-additive analogue of Paley graphs over finite fields and characterizes the structure and clique numbers of these graphs.

Findings

Describes the structure of maximal cliques in the graphs.

Provides bounds on the clique number for arbitrary subspaces.

Calculates exact clique numbers for certain subspace dimensions.

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 $d_U\in \{1, 2, n-1\}$.