Generalized solutions and generalized eigenfunctions of boundary-value problems on a geometric graph

We consider generalized solutions to boundary-value problems for elliptic equations on an arbitrary geometric graph and their corresponding eigenfunctions. We construct analogs of Sobolev spaces that are dense in $L_2$. We obtain conditions for the Fredholm solvability of boundary-value problems of different types, describe their spectral properties and conditions of the decomposition of generalized eigenfunctions. The results presented here are fundamental in the study of boundary control problems of oscillations of multiplex jointed structures, consisting of strings or rods, as well as in the study of cell metabolism.