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

In this paper we study the eigenvalues of laplacian matrices cyclic graphs with one edge weight α and others 1. We denote by n order graph suppose that tends to infinity. notice characteristic polynomial depend only on Re(α). After that, through rest 0<α<1. It is easy see belong [0,4] are asymptotically distributed as function g(x)=4sin2⁡(x/2) [0,π]. obtain a series results about individual behavior eigenvalues. First, describe more precisely their localization in subintervals [0,4]. Second, transform equation form convenient solve numerical methods. particular, prove Newton's method converges for every n≥3. Third, derive asymptotic formulas all eigenvalues, where errors uniformly bounded respect number eigenvalue.