  We elucidate the case in which the Ablowitz-Ladik (AL) type discrete
nonlinear Schr\"Aodinger equa- tion (NLSE) on simple networks (e.g., star
graphs and tree graphs) becomes completely integrable just as in the case of a
simple 1-dimensional (1-d) discrete chain. The strength of cubic nonlinearity
is different from bond to bond, and networks are assumed to have at least two
semi-infinite bonds with one of them working as an incoming bond. The present
work is a nontrivial extension of our preceding one (Sobirov et al, Phys. Rev.
E 81, 066602 (2010)) on the continuum NLSE to the discrete case. We find: (1)
the solution on each bond is a part of the universal (bond-independent) AL
soliton solution on the 1-d discrete chain, but is multiplied by the inverse of
square root of bond-dependent nonlinearity; (2) nonlinearities at individual
bonds around each vertex must satisfy a sum rule; (3) under findings (1) and
(2), there exist an infinite number of constants of motion. As a practical
issue, with use of AL soliton injected through the incoming bond, we obtain
transmission probabilities inversely proportional to the strength of
nonlinearity on the outgoing bonds.
