  As the first step of proving the Hodge-FVH correspondence recently proposed
in [19], we derive the Virasoro constraints and the Dubrovin--Zhang loop
equation for special cubic Hodge integrals. We show that this loop equation has
a unique solution, and provide a new algorithm for the computation of these
Hodge integrals. We also observe the gap phenomenon for certain special cubic
Hodge free energies.
