Distance-integral Cayley graphs over abelian groups and dicyclic groups

Abstract

A graph is said to be distance-integral if every eigenvalue of its distance matrix is an integer. In this paper, we study the distance spectrum of abelian Cayley graphs and a class of non-abelian Cayley graphs, namely Cayley graphs over the dicyclic group $$T_{4n}=\langle a,b\,|\,a^{2n}=1, a^n=b^2, b^{-1}ab=a^{-1}\rangle $$ of order 4n. Based on the representation theory of finite groups, we first show that an abelian Cayley graph is integral if and only if it is distance-integral, which naturally contains a main result obtained in [Electron. J. Comb. 19(4) (2012) paper 25, 8 pp]. Then, we display a necessary and sufficient condition for a Cayley graph over $$T_{4n}$$ to be distance-integral; some simple necessary (or sufficient) conditions for the distance integrality of a Cayley graph over $$T_{4n}$$ in terms of the Boolean algebra of $$\left $$ are provided as well. Consequently, some infinite families of distance-integral Cayley graphs over $$T_{4n}$$ are constructed. Finally, for a prime $$p\ge 3$$ , all the distance-integral Cayley graphs over $$T_{4p}$$ are completely characterized.