This paper characterizes 2-integral Cayley graphs and defines k-integral graphs in finite groups, suggesting finite connected graphs can exist within specified bounds.
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 ℚ is equal to k. We prove that for any positive integers k and Δ, the set of all finite connected graphs with algebraic degree at most k and maximum degree at most Δ is finite. 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 the finite groups G that al Cayley graphs over G with bounded valency are 2-integral.
No takes yet. Share an insight, caveat, or question.
Abdollahi et al. (2025) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: