Algebraic degrees of 2-Cayley digraphs over abelian groups

Abstract

A digraph Γ is called a 2-Cayley digraph over a group G if there exists a 2-orbit semiregular subgroup of Aut(Γ) isomorphic to G. In this paper, we give upper and lower bounds for the algebraic degrees of 2-Cayley digraphs over abelian groups. As an application, we consider the Cayley digraphs over finite groups admitting an abelian subgroup of index 2. Special attention is paid for Cayley graphs over generalized dihedral groups and generalized dicyclic groups. For Cayley graphs over generalized dihedral groups and generalized dicyclic groups, we completely determine their algebraic degrees and splitting fields. At last, the upper and lower bounds for the algebraic degrees of Cayley digraphs over semi-dihedral groups are obtained.