On 2-integral Cayley graphs

Abstract

In this paper, we introduce the concept of k-integral graphs. A graph Γ is called kintegral 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 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. cb This work is licensed under https://creativecommons.org/licenses/by/4.0/