  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 $\operatorname{Re}(\alpha)<0$, and all the others
have weights $1$. This paper is a sequel of a previous one where we considered
$\operatorname{Re}(\alpha) \in[0,1]$ (Eigenvalues of laplacian matrices of the
cycles with one weighted edge, Linear Algebra Appl. 653, 2022, 86--115). We
prove that for $\operatorname{Re}(\alpha)<0$ and
$n>\operatorname{Re}(\alpha-1)/\operatorname{Re}(\alpha)$, one eigenvalue is
negative 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 the half of the inner eigenvalues, while for the others we justify
the convergence of Newton's method and fixed-point iteration method. We find
asymptotic expansions, as $n$ tends to $\infty$, both for the eigenvalues
belonging to $[0,4]$ and the outlier. We also compute the eigenvectors and
their norms.
