A graph is said to be integral (resp. distance integral) if all the eigenvalues of its adjacency matrix (resp. distance matrix) are integers. Let H be a finite abelian group, and let \({\mathscr {H}}=\langle H,b\,|\,b^2=1,bhb=h^{-1},h\in H\rangle \) be the generalized dihedral group of H. The contribution of this paper is threefold. Firstly, based on the representation theory of finite groups, we obtain a necessary and sufficient condition for a Cayley graph over \({\mathscr {H}}\) to be integral, which naturally contains the main results obtained in Lu et al. (J Algebr Comb 47:585–601, 2018). Secondly, a closed-form decomposition formula for the distance matrix of Cayley graphs over any finite groups is derived. As applications, a necessary and sufficient condition for the distance integrality of Cayley graphs over \({\mathscr {H}}\) is displayed. Some simple sufficient (or necessary) conditions for the integrality and distance integrality of Cayley graph are exhibited, respectively, from which several infinite families of integral and distance integral Cayley graphs over \({\mathscr {H}}\) are constructed. And lastly, some necessary and sufficient conditions for the equivalence of integrity and distance integrity of Cayley graphs over generalized dihedral groups are obtained.

如果图的邻接矩阵（分别是距离矩阵）的所有特征值都是整数，则称该图为整数（即​​距离积分）。令H为有限阿贝尔群，令\（{\ mathscr {H}} = \ langle H，b \，| \，b ^ 2 = 1，bhb = h ^ {-1}，h \ in H \ rangle是H的广义二面体组。本文的贡献是三方面的。首先，基于有限群的表示理论，我们获得了\（{\ mathscr {H}} \）上的Cayley图的充要条件是完整的，自然包含了Lu等人获得的主要结果。（J Algebr Comb 47：585–601，2018）。其次，推导了任意有限群上Cayley图距离矩阵的封闭形式分解公式。作为应用，显示了\（{\ mathscr {H}} \）上Cayley图的距离完整性的充要条件。分别展示了一些简单的Cayley图的积分和距离积分的充分（或必要）条件，从中可以得出\（{{mathscr {H}} \}被构造。最后，获得了在广义二面体群上Cayley图的完整性和距离完整性相等的一些必要和充分条件。