Eigenvalues of the laplacian matrices of the cycles with one weighted edge
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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)=4sin^2(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.
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