2022/10/06 by Lucas Reis, Reis, Lucas
Computer Science · Mathematics · #Coding theory and cryptography #Finite Group Theory Research #Cooperative Communication and Network Coding
paper · pdf · doi:10.48550/arxiv.2210.03236
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\subsetneq \mathbb Fqn is an \mathbb Fq-vector space, GU is the (undirected) graph with vertex set V(GU)=\mathbb Fqn and edge set E(GU)=\(a, b)∈ \mathbb 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\subsetneq \mathbb Fqn is any \mathbb Fq-vector space of dimension dU∈ \1, 2, n-1\.