2023/10/23 by Ricardo A. Podestá, Podestá, Ricardo A., Denis E. Videla +1
Engineering · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2310.15378
openalex publication_date 2023/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
We study the spectrum of generalized Paley graphs Γ(k,q)=Cay(\mathbbFq,Rk), undirected or not, with Rk=\xk:x∈ \mathbbFq^*\ where q=pm with p prime and k| q-1. We first show that the eigenvalues of Γ(k,q) are given by the Gaussian periods ηi(k,q) with 0≤ i≤ k-1. Then, we explicitly compute the spectrum of Γ(k,q) with 1≤ k ≤ 4 and of Γ(5,q) for p≡ 1\pmod 5 and 5| m. Also, we characterize those GP-graphs having integral spectrum, showing that Γ(k,q) is integral if and only if p divides (q-1)/(p-1). Next, we focus on the family of semiprimitive GP-graphs. We show that they are integral strongly regular graphs (of pseudo-Latin square type). Finally, we characterize all integral Ramanujan graphs Γ(k,q) with 1≤ k ≤ 4 or where (k,q) is a semiprimitive pair.