EXPLORING CHIP-FIRING GAMES ON EULERIAN AND HERMITIAN GRAPHS APPROACH - Hiremath, Kavita; Kaushal, Namrata; Gupta, Jyoti (ShodhKosh: Journal of Visual and Performing Arts, 2024)
Based on the graph structure and chip distribution on its vertices, three distinct types of dynamic models emerge. Among these, Chip-Firing Games have garnered significant attention due to their wide applicability across various mathematical fields, including algebra, combinatorics, dynamical systems, statistics, algorithms, and computational complexity. In this paper, we explore the characterization properties of configuration spaces, which are organized by predecessor relations. We introduce Chip-Firing Games (CFGs) as a method, termed the "Probability Abacus," to compute absorbing probabilities through vector addition techniques. We demonstrate that the termination state of any CFG within an absorbing Markov chain, characterized by a rational transition matrix, which is independent of the firing order and critical loading reappearing at the termination state.