In this paper, we study spectral properties of the hypercubes, a special kind of Cayley graphs. We determine explicitly all the eigenvalues and their corresponding multiplicities of the normalized Laplacian matrix of the hypercubes by a recursive method. As applications of these results, we derive the explicit formula to the eigentime identity for random walks on the hypercubes and show that it grows linearly with the network order. Moreover, we compute the number of spanning trees and the degree-Kirchhoff index of the hypercubes. Finally, we study the susceptible–infectious–susceptible (SIS) dynamics on the hypercubes and determine the epidemic threshold based on the spectral radius of the adjacency matrix. Throughout this paper, two numerical experiments are conducted based on the dynamics of complex networks, namely, random walks and epidemic spreading, and the results are consistent with our theoretical analysis.

在本文中，我们研究了超立方体（一种特殊的Cayley图）的光谱特性。我们通过递归方法明确确定超立方体的规范化拉普拉斯矩阵的所有特征值及其对应的多重性。作为这些结果的应用，我们推导了超立方体上随机游动的本征时间标识的显式公式，并表明该公式随网络阶数线性增长。此外，我们计算生成树的数量和超立方体的度-基尔霍夫指数。最后，我们研究了超立方体上的易感性-传染性-易感性（SIS）动态，并根据邻接矩阵的光谱半径确定了流行阈值。在本文中，基于复杂网络的动力学进行了两个数值实验，即