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) = 4 sin 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.
In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries 2, −1, 0, . . . , 0, −α in the first column. Notice that the generating symbol depends on the order n of the matrix. If |α| ≤ 1, then the eigenvalues belong to [0, 4] and are asymptotically distributed as the function g(x) = 4 sin 2 (x/2) on [0, π]. The situation changes drastically when |α| > 1 and n tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of [0, 4] and converge rapidly to certain limits determined by the value of α, whilst all others belong to [0, 4] and are asymptotically distributed as g. In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues.
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