论文标题
$ n $ -polygon图的标准化拉普拉斯频谱及其应用
The normalized Laplacian spectrum of $n$-polygon graphs and its applications
论文作者
论文摘要
给定任意连接的$ g $，通过在图形$ g $的每个边缘添加长度$ n $ $（n \ geq 2）$获得的$ n $ -polygon图$τ_n（g）$，获得$ n $ g $的路径，以及迭代$ n $ n $ n $ -polygon $ -polygon $ them-polygon $τ_n^g（g）$ g（g） $τ_n^g（g）=τ_n（τ_n^{g-1}（g））$，初始条件$τ_n^0（g）= g $。在本文中，如果首先给出了图形$τ_n（g）$的归一化拉普拉斯矩阵的特征值的方法，则如果首先给出了图形$ g $的归一化laplacian矩阵的特征值。然后，也可以得出图形$τ_n（g）$的标准化拉普拉斯频谱和图形$τ_n^g（g）$（$ g \ geq 0 $）。最后，作为应用程序，我们通过探索与归一化的laplacian Spect的连接来探索他们的连接，从而计算出图$τ_n（g）$的乘法kirchhoff索引，凯门尼的常数以及图$τ_n（g）$的跨越树的数量，并获得了这些量子的精确结果。
Given an arbitrary connected $G$, the $n$-polygon graph $τ_n(G)$ is obtained by adding a path with length $n$ $(n\geq 2)$ to each edge of graph $G$, and the iterated $n$-polygon graphs $τ_n^g(G)$ ($g\geq 0$), is obtained from the iteration $τ_n^g(G)=τ_n(τ_n^{g-1}(G))$, with initial condition $τ_n^0(G)=G$. In this paper, a method for calculating the eigenvalues of normalized Laplacian matrix for graph $τ_n(G)$ is presented if the eigenvalues of normalized Laplacian matrix for graph $G$ is given firstly. Then, the normalized Laplacian spectrums for the graph $τ_n(G)$ and the graphs $τ_n^g(G)$ ($g\geq 0$) can also be derived. Finally, as applications, we calculate the multiplicative degree-Kirchhoff index, Kemeny's constant and the number of spanning trees for the graph $τ_n(G)$ and the graphs $τ_n^g(G)$ by exploring their connections with the normalized Laplacian spectrum, exact results for these quantities are obtained.
