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 α∈C, with Re(α)<0, and all the others have weights 1. This paper is a sequel of a previous one where we considered Re(α)∈[0,1] (Grudsky et al., 2022 [12]). We prove that for Re(α)<0 and n>Re(α−1)/Re(α), one eigenvalue is negative while the others belong to [0,4] and are distributed as the function x↦4sin2⁡(x/2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4Re(α)2/(2Re(α)−1). We give exact formulas for 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 ∞, both for the eigenvalues belonging to [0,4] and the outlier. We also compute the eigenvectors and their norms.