  The present work introduces two possible extensions of the Standard Model
Higgs sector. In the first case, the Zee-Babu type model for the generation of
neutrino mass is augmented with a scalar triplet and additional singly charged
scalar singlets. The second scenario, on the other hand, generalizes the
Type-II seesaw model by replicating the number of the scalar triplets. A
$\mathbb{Z}_3$ symmetry is imposed in case of both the scenarios, but, allowed
to be violated by terms of mass dimension two and three for generating neutrino
masses and mixings. We examine how the models so introduced can explain the
experimental observation on the muon anomalous magnetic moment. We estimate the
two-loop contribution to neutrino mass induced by the scalar triplet, in
addition to what comes from the doubly charged singlet in the usual Zee-Babu
framework, in the first model. On the other hand, the neutrino mass arises in
the usual Type-II fashion in the second model. In addition, the role of the
$\mathbb{Z}_3$ symmetry in suppressing lepton flavor violation is also
elucidated.
