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

TL;DR

This paper investigates the eigenvalues of Laplacian matrices of cyclic graphs with one weighted edge, providing asymptotic formulas, localization results, and numerical solution methods, especially for large graphs.

Contribution

It offers new asymptotic formulas and localization results for eigenvalues of Laplacian matrices of cycles with one weighted edge, and proves convergence of numerical methods.

Findings

Eigenvalues are in [0,4] and distributed as 4sin^2(x/2) for large n

Localization of eigenvalues in specific subintervals

Newton's method converges for all n ≥ 3

Abstract

In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight $\alpha$ 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 $\operatorname{Re}(\alpha)$. After that, through the rest of the paper we suppose that $0<\alpha<1$. 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 $[0,\pi]$. 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.