  Laser irradiation, as a versatile tool to tune topological properties of
electronic systems, is under intensive studies. Experimentally, laser
irradiation induced anomalous Hall effect in graphene has been observed (McIver
et al., Nat. Phys. 16, 38 (2020)). Disorder is ubiquitous in real materials,
and it has been shown that diagonal disorders, i.e., onsite disorder, can
enhance topological properties of time-periodically driven quantum materials
(Titum et al., Phys. Rev. Lett. 114, 056801 (2015)). Here, we investigate
circularly polarized laser irradiated graphene with non-diagonal disorders,
i.e., disordered tunneling, and find that disorder can induce nontrivial
topological properties, characterized by Bott index and the real-space Chern
number. Moreover, we show that one can turn on the laser irradiation
non-adiabatically to drive the disordered graphene into non-trivial topological
phase. It is a scheme which is especially interesting for experimental
implementations.
