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

TL;DR

This paper analyzes the eigenvalues of Laplacian matrices of cyclic graphs with one negatively weighted edge, revealing eigenvalue distribution, asymptotic behavior, and convergence properties as the graph size grows.

Contribution

It provides new explicit formulas, asymptotic expansions, and convergence results for eigenvalues of Laplacian matrices with negative edge weights, extending previous work.

Findings

One eigenvalue becomes negative for large n.

Remaining eigenvalues are within [0,4] and follow a sinusoidal distribution.

The outlier eigenvalue converges exponentially to a specific limit.

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 $\alpha\in\mathbb{C}$, with $\operatorname{Re}(\alpha)<0$, and all the others have weights $1$. This paper is a sequel of a previous one where we considered $\operatorname{Re}(\alpha) \in[0,1]$ (Eigenvalues of laplacian matrices of the cycles with one weighted edge, Linear Algebra Appl. 653, 2022, 86--115). We prove that for $\operatorname{Re}(\alpha)<0$ and $n>\operatorname{Re}(\alpha-1)/\operatorname{Re}(\alpha)$, 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 $4\operatorname{Re}(\alpha)^2/(2\operatorname{Re}(\alpha)-1)$. 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.