Chip-firing and energy minimization on M-matrices

@article{Guzmn2015ChipfiringAE,
  title={Chip-firing and energy minimization on M-matrices},
  author={J. Guzm{\'a}n and Caroline J. Klivans},
  journal={J. Comb. Theory, Ser. A},
  year={2015},
  volume={132},
  pages={14-31}
}
  • J. Guzmán, Caroline J. Klivans
  • Published 2015
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • We consider chip-firing dynamics defined by arbitrary M-matrices. M-matrices generalize graph Laplacians and were shown by Gabrielov to yield avalanche finite systems. Building on the work of Baker and Shokrieh, we extend the concept of energy minimizing chip configurations. Given an M-matrix, we show that there exists a unique energy minimizing configuration in each equivalence class defined by the matrix.We consider the class of z-superstable configurations. We prove that for any M-matrix… CONTINUE READING
    23 Citations

    Topics from this paper

    A maximizing characteristic for critical configurations of chip-firing games on digraphs
    • PDF
    Cycle and circuit chip-firing on graphs
    • Highly Influenced
    • PDF
    Chip firing on Dynkin diagrams and McKay quivers
    • 15
    • PDF
    A chip-firing variation and a Markov chain with uniform stationary distribution
    • 1
    • PDF
    Abelian networks IV. Dynamics of nonhalting networks
    • 1
    • PDF
    G-Strongly Positive Scripts and Critical Configurations of Chip Firing Games on Digraphs
    • Tran Thi Thu Huong
    • Mathematics, Computer Science
    • 2016 International Conference on Advanced Computing and Applications (ACOMP)
    • 2016
    • 1
    • PDF
    Counting arithmetical structures on paths and cycles
    • 10
    • PDF
    Uniform Bounds for Non-negativity of the Diffusion Game.
    • 2
    • PDF

    References

    SHOWING 1-10 OF 31 REFERENCES
    Chip-Firing and Riemann-Roch Theory for Directed Graphs
    • 25
    • PDF
    Chip-firing games, potential theory on graphs, and spanning trees
    • 86
    • Highly Influential
    • PDF
    Chip-firing Games on Graphs
    • 270
    • PDF
    The Chip Firing Game and Matroid Complexes
    • C. Merino
    • Mathematics, Computer Science
    • DM-CCG
    • 2001
    • 42
    • PDF
    Asymmetric Abelian Avalanches and Sandpiles
    • 14
    • Highly Influential
    • PDF
    G-parking functions, acyclic orientations and spanning trees
    • 48
    • PDF
    PRIMER FOR THE ALGEBRAIC GEOMETRY OF SANDPILES
    • 61
    • Highly Influential
    • PDF
    Self-organized critical state of sandpile automaton models.
    • Dhar
    • Mathematics, Medicine
    • Physical review letters
    • 1990
    • 665
    Chip-Firing and the Critical Group of a Graph
    • 256
    • PDF