We show that under current experimental bounds of the decays ${e}_{a}\ensuremath{\rightarrow}{e}_{b}\ensuremath{\gamma}$, the recent experimental data of the muon anomalous magnetic dipole moment $(g\ensuremath{-}2{)}_{\ensuremath{\mu}}$ can be explained in the framework of the $3\ensuremath{-}3\ensuremath{-}1$ model with right-handed neutrinos. In addition, all of these branching ratios can reach closely the recent experimental upper bounds.