Parameter dependent differential operators on graphs and their applications

The paper is devoted to parameter dependent differential operators on graphs with general vertex conditions. We prove that if the differential operators are elliptic with parameter and an analogue of the well-known Lopatinsky condition at the vertices is satisfied, then the parameter dependent problem on the graph is invertible for large values of parameters. We consider applications of these results to the invertibility of general parabolic initial-vertex value problems on graphs, in particular case, for the heat equations on graphs with variable coefficients and general vertex conditions.