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

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 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 >= 3. Third, we derive asymptotic formulas for all eigenvalues, where the errors are uniformly bounded with respect to the number of the eigenvalue. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we investigate the eigenvalues of the laplacian matrices of cyclic graphs with one edge of weight alpha and the others of weight 1. We find that the characteristic polynomial and eigenvalues depend only on Re(alpha). We also study the individual behavior of the eigenvalues, including their localization, numerical solution of the characteristic equation, and asymptotic formulas.