In this paper vertex conditions for the fourth order differential operator on the simplest metric graph - the $ Y$-graph, -are discussed. In order to make the operator symmetric one needs to impose extra conditions on the limit values offunctions and their derivatives at the central vertex. It is shown that such conditions corresponding to the free movement of beams depend on the angles between the beams in the equilibrium position.

In this paper vertex conditions for the differential operator of fourth derivative on the simplest metric graph - the Y-graph, - are discussed. In order to make the operator symmetric one needs to impose extra conditions on the limit values of functions and their derivatives at the central vertex. It is shown that such conditions corresponding to the free movement of beams depend on the angles between the beams in the equilibrium position.

In the current article it is analyzed how ideas of PT-symmetricquantum mechanics can be applied to quantum graphs, in particular tothe star graph. The class of rotationally-symmetric vertex conditionsis analyzed. It is shown that all such conditions can effectively be described bycirculant  matrices: real in the case of odd number of edges and complex having particular block structure  in the even case. Spectral properties of thecorresponding operators are discussed.