ω-Harmonic Functions and Inverse Conductivity Problems on Networks

In this paper, we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network, in terms of an input-output map. To do this we deal with the weighted Laplacian$\Delta_{\omega}$and an$\omega-harmonic$function on the graph, with its physical interpretation as a diffusion equation on the graph, which models an electric network. After deriving the basic properties of$\omega-harmonic$functions, we prove the solvability of (direct) problems such as the Dirichlet and Neumann BVPs. Our main result is the global uniqueness of the inverse conductivity problem for a network under a suitable monotonicity condition.