We consider quantum graphs with transparent branching points. To design such networks, the concept of transparent boundary conditions is applied to the derivation of the vertex boundary conditions for the linear Schrödinger equation on metric graphs. This allows to derive simple constraints, which use equivalent usual Kirchhoff-type boundary conditions at the vertex to the transparent ones. The approach is applied to quantum star and tree graphs. However, extension to more complicated graph topologies is rather straight forward.

We consider Bogoliubov de Gennes equation on metric graphs. The vertex boundary conditions providing self-adjoint realization of the Bogoliubov de Gennes operator on a metric star graph are derived. Secular equation providing quantization of the energy and the vertex transmission matrix are also obtained. Application of the model for Majorana wire networks is discussed.

In this paper, we study quantum star graphs with time-dependent bond lengths. Quantum dynamics are treated by solving Schrodinger equation with time-dependent boundary conditions given on graphs. The time-dependence of the average kinetic energy is analyzed. The space-time evolution of a Gaussian wave packet is treated for an harmonically breathing star graph.

We consider the reflectionless transport of solitons in networks. The system is modeled in terms of the nonlinear Schrödinger equation on metric graphs, for which transparent boundary conditions at the branching points are imposed. This allows to derive simple constraints, which link equivalent usual Kirchhoff-type vertex conditions to the transparent ones. Our approach is applied to a metric star graph. An extension to more complicated graph topologies is straight forward.

We study the quantum dynamics of Gaussian wave packets on star graphs whose arms feature each a periodic potential and an external time-dependent field. Assuming that the potentials and the field can be manipulated separately for each arm of the star, we show that it is possible to manipulate the direction of the motion of a Gaussian wave packet through the bifurcation point by a suitable choice of the parameters of the external fields. In doing so, one can achieve a transmission of the wave packet into the desired arm with nearly 70% while also keeping the shape of the wave packet approximately intact. Since a star graph is the simplest element of many other complex graphs, the obtained results can be considered as the first step to wave packet manipulations on complex networks.