Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge

TL;DR

This work analyzes the eigenvalues of the Laplacian of a cycle with one edge weighted by $\alpha$ where $\operatorname{Re}(\alpha)<0$. By exploiting a Chebyshev-polynomial representation of the characteristic polynomial, the authors localize eigenvalues, showing one negative outlier and the rest confined to $[0,4]$ with the familiar $g(x)=4\sin^2(x/2)$ distribution; the outlier converges exponentially to $\Omega_\alpha=4\alpha^2/(2\alpha-1)$. They derive explicit equations for inner eigenvalues, establish fixed-point and Newton methods with rigorous convergence for large $n$, and obtain detailed asymptotic expansions for both inner and outlier eigenvalues, along with explicit eigenvectors and their norms. Numerical experiments confirm the asymptotics and the efficiency of the computational schemes. The results extend previous work on $\operatorname{Re}(\alpha)\in[0,1]$ to $\operatorname{Re}(\alpha)<0$, providing a comprehensive spectral picture for this structured matrix family and its implications for heat and wave propagation on the associated graphs.

Abstract

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order $n$, where one edge has weight $α\in\mathbb{C}$, with $\operatorname{Re}(α)<0$, and all the others have weights $1$. This paper is a sequel of a previous one where we considered $\operatorname{Re}(α) \in[0,1]$ (Eigenvalues of laplacian matrices of the cycles with one weighted edge, Linear Algebra Appl. 653, 2022, 86--115). We prove that for $\operatorname{Re}(α)<0$ and $n>\operatorname{Re}(α-1)/\operatorname{Re}(α)$, one eigenvalue is negative while the others belong to $[0,4]$ and are distributed as the function $x\mapsto 4\sin^2(x/2)$. Additionally, we prove that as $n$ tends to $\infty$, the outlier eigenvalue converges exponentially to $4\operatorname{Re}(α)^2/(2\operatorname{Re}(α)-1)$. We give exact formulas for the half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as $n$ tends to $\infty$, both for the eigenvalues belonging to $[0,4]$ and the outlier. We also compute the eigenvectors and their norms.

Figures (8)

• Figure 1: Graph $G_{\alpha,8}$
• Figure 2: Plot of $g$ (blue), plot of $x\mapsto g_-(-x)$ (green), points $z_{\alpha,n,j}$ and $s_{\alpha,n}$, and the corresponding values of $\lambda_{\alpha,n,j}$, for $\alpha = -1/2$ and $n= 8$. The red labels on the horizontal axis are $j\pi/n$.
• Figure 3: Plot of $\eta_\alpha$ (blue) and the left-hand side of \ref{['eq:equation_with_eta']} (green) for $\alpha = -1/2$, $n=8$ (left) and $\alpha=-3$, $n=9$ (right).
• Figure 4: Plots of the functions from Theorem \ref{['thm:contraction_inner_left']} and their fixed points, for $\alpha = -1/2$, $n=8$ (left) and $\alpha=-3$, $n=9$ (right).
• Figure 5: Plot of $\varphi_{\alpha,n}$ (blue), tangent line to the graph of $\varphi_{\alpha,n}$ at $\ell_{\alpha,n}$ (purple), and plot of $x\mapsto x$ (green), for $\alpha = -1/2$ and $n= 6$.

Theorems & Definitions (54)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Theorem 2.1: eigenvalues' localization
• Theorem 2.2: main equations
• Theorem 2.3: asymptotic expansion of inner eigenvalues
• Theorem 2.4: asymptotic expansion of the first eigenvalue
• Theorem 2.5: eigenvectors for $\operatorname{Re}(\alpha)<0$
• Theorem 2.6: norms of eigenvectors for $\operatorname{Re}(\alpha)<0$
• Proposition 3.1: characteristic polynomial of $L_{\alpha,n}$ for complex $\alpha$