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 \(\left<a\right>\) 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.

如果图的距离矩阵的每个特征值都是整数，则称该图为距离积分。本文研究了Abelian Cayley图和一类非Abelian Cayley图的距离谱，即双环群\（T_ {4n} = \ langle a，b \，| \，a ^ { 2n} = 1，a ^ n = b ^ 2，b ^ {-1} ab = a ^ {-1} \ rangle \）的阶数为4 n。基于有限群的表示理论，我们首先证明abelian Cayley图是且仅当它是距离积分时才是积分的，它自然包含[电子学]中得到的主要结果。J.梳 19（4）（2012）25，8 pp]。然后，我们显示\（T_ {4n} \）上的Cayley图的充要条件距离积分；还提供了根据\（\ left <a \ right> \）的布尔代数，关于\（T_ {4n} \）上Cayley图的距离积分的一些简单必要（或充分）条件。因此，构造了\（T_ {4n} \）上的一些无穷远距离的Cayley图族。最后，对于素数\（p \ ge 3 \），\（T_ {4p} \）上的所有距离积分Cayley图都得到了完全刻画。