Perturbed invariant under a cyclic permutation with trace of neutrino mass matrix remain constant

We construct a neutrino mass matrix $M_{\nu}$ via a seesaw mechanism whith perturbed invariant under a cyclic permutation by introducing one parameter $\delta$ into the diagonal elements of $M_{\nu}$ with assumption that trace of the perturbed $M_{\nu}$ is equal to trace of the unperturbed $M_{\nu}$. We found that the perturbed neutrino mass matrices $M_{\nu}$ can predicts the mass-squared difference $\Delta m_{ij}^{2}\neq 0$ with the possible hierarchy of neutrino mass is normal or inverted hierarchy. By using the advantages of the mass-squared differences and mixing parameters data from neutrino oscillaton experiments, we obtained neutrino masses in inverted hierarchy with masses: $|m_{1}|=0.101023$ eV, $|m_{2}|=0.101428$ eV, and $|m_{3}|=0.084413$ eV.