Integral Flow and Cycle Chip-Firing on Graphs

Motivated by the notion of chip-firing on dual graph a planar graph, we consider ‘integral flow chip-firing’ an arbitrary G. The rule is governed $${\mathcal {L}}^*(G)$$ , Laplacian G determined choosing basis for lattice integral flows We show that any admits such so M-matrix, leading to firing these elements avalanche finite. This follows from more general result bases lattices may be independent interest. Our results provide z-superstable configurations are in bijection with set spanning trees graphs, as well graphs $$K_5$$ and $$K_{3,3}$$ one can find M-basis consists cycles underlying graph. question address some open questions.