Elliptic differential operators on infinite graphs with general conditions on vertices

Let Γ be an unbounded metric-oriented graph embedded in (Formula presented.), (Formula presented.), (Formula presented.) be countable sets of edges (Formula presented.), and vertices (Formula presented.) of Γ. The graph Γ is equipped with a differential equation (Formula presented.) with piece-wise smooth coefficients (Formula presented.) (Formula presented.), and general conditions at the vertices (Formula presented.) (Formula presented.) where (Formula presented.) is the number of edges incident to v, (Formula presented.) are (Formula presented.) complex matrices, (Formula presented.) (Formula presented.) are limit values at the vertex (Formula presented.) of the derivatives (Formula presented.) taken along the edges (Formula presented.) according to their orientation. We associate with equation (1) and the vertex conditions (2) an operator (Formula presented.) acting from the Sobolev space (Formula presented.) to the space (Formula presented.) where (Formula presented.). We study the smoothness, and exponential behavior at infinity of solutions of equation (Formula presented.), and for periodic graphs we obtain the necessary and sufficient conditions of the Fredholmness of (Formula presented.) and a description of the essential spectrum of the realization of (Formula presented.) in (Formula presented.) We give applications of these results to the Schrödinger operators on periodic graphs with general conditions at the vertices.