  In this paper, we discuss the concept of quantum graphs with transparent
vertices by considering the case where the graph interacts with an external
time-independent field. In particular, we address the problem of transparent
boundary conditions for quantum graphs, building on previous work on
transparent boundary conditions for the stationary Schrodinger equation on a
line. Physically relevant constraints making the vertex transparent under
boundary conditions in the form of (weight) continuity and Kirchhoff rules are
derived using two methods, the scattering approach and transparent boundary
conditions for the time-independent Schrodinger equation. The latter is derived
by extending the transparent boundary condition concept to the time-independent
Schrodinger equation on driven quantum graphs. We also discuss how the
eigenvalues and eigenfunctions of a quantum graph are influenced not only by
its topology, but also by the shape(type) of a potential when an external field
is involved.
