  In this work, we construct and quantify asymptotically in the limit of large
mass a variety of edge-localized stationary states of the focusing nonlinear
Schr\"odinger equation on a quantum graph. The method is applicable to general
bounded and unbounded graphs. The solutions are constructed by matching a
localized large amplitude elliptic function on a single edge with an
exponentially smaller remainder on the rest of the graph. This is done by
studying the intersections of Dirichlet-to-Neumann manifolds (nonlinear
analogues of Dirichlet-to-Neumann maps) corresponding to the two parts of the
graph. For the quantum graph with a given set of pendant, looping, and internal
edges, we find the edge on which the state of smallest energy at fixed mass is
localized. Numerical studies of several examples are used to illustrate the
analytical results.
