论文标题
一个加权边缘的循环的拉普拉斯矩阵的特征值
Eigenvalues of the laplacian matrices of the cycles with one weighted edge
论文作者
论文摘要
在本文中，我们研究了一个重量$α$的循环图的拉普拉斯矩阵的特征值和重量$ 1 $的特征值。我们用$ n $表示图表的顺序，并假设$ n $倾向于无限。我们注意到，特征多项式和特征值仅取决于$ \ operatatorName {re}（α）$。之后，通过纸张的其余部分，我们假设$ 0 <α<1 $。很容易看出特征值属于$ [0,4] $，并且在$ [0，π] $上的函数$ g（x）= 4 \ sin^2（x/2）$中渐近分布。我们获得了有关特征值的个体行为的一系列结果。首先，我们更精确地描述了它们在$ [0,4] $的子间隔中的本地化。其次，我们将特征方程式转换为方便地通过数值方法求解的形式。特别是，我们证明牛顿的方法每$ n \ ge3 $都收敛。第三，我们为所有特征值得出渐近公式，其中误差相对于特征值的数量均匀地界定。
In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight $α$ and the others of weight $1$. We denote by $n$ the order of the graph and suppose that $n$ tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on $\operatorname{Re}(α)$. After that, through the rest of the paper we suppose that $0<α<1$. It is easy to see that the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,π]$. We obtain a series of results about the individual behavior of the eigenvalues. First, we describe more precisely their localization in subintervals of $[0,4]$. Second, we transform the characteristic equation to a form convenient to solve by numerical methods. In particular, we prove that Newton's method converges for every $n\ge3$. Third, we derive asymptotic formulas for all eigenvalues, where the errors are uniformly bounded with respect to the number of the eigenvalue.
