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Hyperbolic systems on networks often can be written as systems of first orderequations on an interval, coupled by transmission conditions at the endpoints,also called port-Hamiltonians. However, general results for the latter havebeen difficult to interpret in the network language. The aim of this paper isto derive conditions under which a port-Hamiltonian with general linearKirchhoff's boundary conditions can be written as a system of $2\times 2$hyperbolic equations on a metric graph $\Gamma$. This is achieved byinterpreting the matrix of the boundary conditions as a potential map of vertexconnections of $\Gamma$ and then showing that, under the derived assumptions,that matrix can be used to determine the adjacency matrix of $\Gamma$.

Telegraph type systems on networks and port-Hamiltonians. II. Graph realizability