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

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.