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

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

28 pages, 8 figures

References in corpus (1)

• Eigenvalues of the laplacian matrices of the cycles with one weighted edge