The Kirchhoff index of unicycle weighted chains

We consider a class of generalized linear chains, namely, the unicycle chains as a perturbation of a 2n–path by adding one weighted edge between two specific vertices. This class of chains includes, in particular, the weighted cycle. In this work, we obtain the Green function, the effective resistance and the Kirchhoff index of the unicycle chains as a function of the Green function, the effective resistance and the Kirchhoff index of a path, respectively.