- The delta interaction at a vertex generalizes the Robin (generalized Neumann) boundary condition on an interval. Study of a single vertex with N infinite leads suffices to determine the localized effects of such a vertex on densities of states, etc. For all the standard initial-value problems, such as that for the wave equation, the pertinent integral kernel (Green function) on the graph can be easily constructed from the corresponding elementary Green function on the real line. From the results one obtains the spectral-projection kernel, local spectral density, and local energy density. The energy density, which refers to an interpretation of the graph as the domain of a quantized scalar field, is a coefficient in the asymptotic expansion of the Green function for an elliptic problem involving the graph Hamiltonian; that expansion contains spectral/geometrical information beyond that in the much-studied heat-kernel expansion.