TY - JOUR
T1 - Eigenvalues of the laplacian matrices of the cycles with one weighted edge
AU - Grudsky, Sergei M.
AU - Maximenko, Egor A.
AU - Soto-González, Alejandro
N1 - Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/11/15
Y1 - 2022/11/15
N2 - In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight α 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(α). After that, through the rest of the paper we suppose that 0<α<1. It is easy to see that the eigenvalues belong to [0,4] and are asymptotically distributed as the function g(x)=4sin2(x/2) on [0,π]. 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.
AB - In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight α 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(α). After that, through the rest of the paper we suppose that 0<α<1. It is easy to see that the eigenvalues belong to [0,4] and are asymptotically distributed as the function g(x)=4sin2(x/2) on [0,π]. 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.
KW - Asymptotic expansion
KW - Eigenvalue
KW - Laplacian matrix
KW - Perturbation
KW - Toeplitz matrix
KW - Weighted cycle
UR - http://www.scopus.com/inward/record.url?scp=85135908327&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2022.07.011
DO - 10.1016/j.laa.2022.07.011
M3 - Artículo
AN - SCOPUS:85135908327
SN - 0024-3795
VL - 653
SP - 86
EP - 115
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
ER -