2024/01/27 by Aliréza Abdollahi, Majid Arezoomand, Abdollahi, Alireza +5
Engineering · Mathematics · #05C25 #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2401.15306
openalex publication_date 2024/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we introduce the concept of k-integral graphs. A graph Γ is called k-integral if the extension degree of the splitting field of the characteristic polynomial of Γ over rational field \mathbb Q is equal to k. We prove that the set of all finite connected graphs with given algebraic degree and maximum degree is finite. 1-integral graphs are just integral ones, graphs all of whose eigenvalues are integer. We study 2-integral Cayley graphs over finite groups G with respect to Cayley sets which are a union of conjugacy classes of G. Among other general results, we completely characterize all finite abelian groups having a connected 2-integral Cayley graph with valency 2,3,4 and 5. Furthermore, we classify finite groups G for which all Cayley graphs over G with bounded valency are 2-integral.