  We show soliton solutions of nonlinear Schroedinger equation on simple
networks consisting of vertices and bonds, where the strength of cubic
nonlinearity is different from bond to bond. We concentrate on reflectionless
propagation of Zakharov-Shabat's solitons through a branched chain, namely, a
primary star graph consisting of three semi-infinite bonds connected at a
vertex. The conservation of the norm and the global current elucidates: (1) the
solution on each bond is a part of the universal soliton solution on a simple
1-dimensional (1-d) chain but multiplied by the inverse of square root of
bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each
vertex must satisfy a sum rule. Under these conditions, all other conservation
rules for a simple 1-d chain have proved to hold for multi-soliton solutions on
graphs. The argument is extended to other graphs, i.e., general star graphs,
tree graphs, loop graphs and their combinations. Numerical evidence is also
given on the reflectionless propagation of a soliton through a branched chain.
