The green function of the boundary-value problem on a star-shaped graph

We consider a planar graph consisting of three edges with one common vertex. We are interested in the sign of the Green function of the boundary-value problem for a fourth-order equation. This problem models deformations of star-shaped coupled networks of beams. We assume that the network is fixed at each vertex, and all beams are rigidly jointed at their common vertex. We prove that the Green function is positive on diagonal squares and establish a sufficient condition for its positivity inside its definition domain.