We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n, where one edge has weight \(\alpha \in \mathbb {C}\) , with \(\operatorname {Re}(\alpha )>1\) , and all the others have weights 1. This paper is a sequel to two previous ones where we considered \(\operatorname {Re}(\alpha ) \in [0,1]\) and \(\operatorname {Re}(\alpha )<0\) . Now, we prove that for \(\operatorname {Re}(\alpha )>1\) and \(n > \operatorname {Re}(\alpha )/\operatorname {Re}(\alpha -1)\) , one eigenvalue is greater than 4 while the others belong to \([0,4]\) and are distributed as the function \(x\mapsto 4\sin ^2(x/2)\) . Additionally, we prove that as n tends to \(\infty \) , the outlier eigenvalue converges exponentially to \(4\operatorname {Re}(\alpha )^2/(2\operatorname {Re}(\alpha )-1)\) . We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton’s method and the fixed-point iteration method. We find asymptotic expansions, as n tends to \(\infty \) , both for the eigenvalues belonging to \([0,4]\) and the outliers. We also compute the eigenvectors and their norms.