Paley-like graphs over finite fields from vector spaces

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≥2 and U⊊Fqn is an Fq-vector space, GU is the (undirected) graph with vertex set V(GU)=Fqn and edge set E(GU)={(a,b)∈Fqn2|a≠b,ab∈U}. We describe the structure of an arbitrary maximal clique in GU and provide bounds on the clique number ω(GU) of GU. In particular, we compute the largest possible value of ω(GU) for arbitrary q and n. Moreover, we obtain the exact value of ω(GU) when U⊊Fqn is any Fq-vector space of dimension dU∈{1,2,n−1}.