We define "BPS graphs" on punctured Riemann surfaces associated with
AN−1 theories of class S. BPS graphs provide a bridge between
two powerful frameworks for studying the spectrum of BPS states: spectral
networks and BPS quivers. They arise from degenerate spectral networks at
maximal intersections of walls of marginal stability on the Coulomb branch.
While the BPS spectrum is ill-defined at such intersections, a BPS graph
captures a useful basis of elementary BPS states. The topology of a BPS graph
encodes a BPS quiver, even for higher-rank theories and for theories with
certain partial punctures. BPS graphs lead to a geometric realization of the
combinatorics of Fock-Goncharov N-triangulations and generalize them in
several ways.Comment: 48 pages, 44 figure