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In this work, we construct and quantify asymptotically in the limit of largemass a variety of edge-localized stationary states of the focusing nonlinearSchr\"odinger equation on a quantum graph. The method is applicable to generalbounded and unbounded graphs. The solutions are constructed by matching alocalized large amplitude elliptic function on a single edge with anexponentially smaller remainder on the rest of the graph. This is done bystudying the intersections of Dirichlet-to-Neumann manifolds (nonlinearanalogues of Dirichlet-to-Neumann maps) corresponding to the two parts of thegraph. For the quantum graph with a given set of pendant, looping, and internaledges, we find the edge on which the state of smallest energy at fixed mass islocalized. Numerical studies of several examples are used to illustrate theanalytical results.

Edge-localized states on quantum graphs in the limit of large mass