A network is defined as an abstract structure that consists of nodes that are connected by links. In this paper, we study two types of rounded networks ${\mathbb{N}}_n$ (resp., ${\mathbb{N}}_n^{\prime }$). By using the recursive relation, we obtain all the eigenvalues and their multiplicities with regards to the associated Laplacian matrix. Meanwhile, we utilize the corresponding relationship between the roots and the coefficients of the characteristic polynomial. Based on these relationships, we obtain the analytical expressions for the sum of the reciprocals of all nonzero Laplacian eigenvalues. By the decomposition theorem of Laplacian polynomial, we obtained an explicit closed-form formula of the Kirchhoff index for ${\mathbb{N}}_n$. As an application of the Laplacian spectra, we reduce the number of spanning trees and the global mean-first passage time for ${\mathbb{N}}_n$. Furthermore, we show that the Kirchhoff index of ${\mathbb{N}}_n$ is approximately to $\frac{1}{3}$ of its Wiener index. In view of our obtained results, all the corresponding results are considered for ${\mathbb{N}}_n^{\prime }$.

网络被定义为由通过链接连接的节点组成的抽象结构。在本文中，我们研究了两种类型的圆形网络${\mathbb{ñ}}_n$（分别，${\mathbb{ñ}}_n^{\text{'}}$）。通过使用递归关系，我们获得了关于相关拉普拉斯矩阵的所有特征值及其多重性。同时，我们利用了特征多项式的根与系数之间的对应关系。基于这些关系，我们得到了所有非零拉普拉斯特征值的倒数之和的解析表达式。通过拉普拉斯多项式的分解定理，我们得到了基尔霍夫指数的一个显式闭式公式：${\mathbb{ñ}}_n$. 作为拉普拉斯谱的应用，我们减少了生成树的数量和全局平均首次通过时间${\mathbb{ñ}}_n$. 此外，我们证明了基尔霍夫指数${\mathbb{ñ}}_n$大约是$\frac{1}{3}$其维纳指数。鉴于我们获得的结果，所有相应的结果都被认为是${\mathbb{ñ}}_n^{\text{'}}$.