In this thesis we focus on the investigation of the relation between the the chip-firing and fractional coloring. Since chi_{f}(G)=inf {n/k : G is homomorphic to K(n,k)}, we find that G has an (n,k)-periodic sequence for some configuration if and only if G is homomorphic to K(n,k). Then we study the periodic configurations for the Kneser graphs. Finally, we try to evaluate the number of chips of the periodic configurations for K(n,k).

We study two deterministic analogues of random walks. The first is the chip-firing game, a single player game played by moving chips around a directed graph, popularised by Björner and Lovász. We find an efficient simulation of boolean circuits and Turing machines using instances of the chip-firing game - after assigning a fixed strategy to the player. The second is the Propp machine, or the rotor router model, a quasirandom model intro- duced by Priezzhev. We improve results of Kijima et al. and show a bound of O(m) on the discrepancy of this process from a random walk on d-regular graphs with m edges. 1

Supersymmetry is a theoretical model of particle physics that posits a symmetry between bosons and fermions. Supersymmetry proposes the existence of particles that we have not yet observed and through them, offers a more unified view of the universe. In the same way Feynman Diagrams represent Feynman Integrals describing subatomic particle behaviour, supersymmetry algebras can be represented by graphs called adinkras. In addition to being motivated by physics, these graphs are highly structured and mathematically interesting. No one has looked at the Jacobians of these graphs before, so we attempt to characterize them in this thesis. We compute Jacobians through the 11-cube, but do not discover any significant discernible patterns. We then dedicate the rest of our work to generalizing the notion of the Jacobian, specifically to be sensitive to edge directions. We conclude with a conjecture describing the form of the directed Jacobian of the directed $n$-topology. We hope for this work to be useful for theoretical particle physics and for graph theory in general.

The intent of this thesis is to explore whether any patterns emerge among families or through graph operations regarding the appearance of Weierstrass vertices on graphs. Currently, patterns have been identified and proven on cycles, complete graphs, complete bipartite graphs, and the house and house-x graphs. A Python program developed as part of this thesis to perform the algorithms used in this analysis confirms these findings. This program also revealed a pattern: if v is a Weierstrass vertex, then the vertex v* added to the graph as a pendant vertex to v is also a Weierstrass vertex. The converse is also true: if v is not a Weierstrass vertex, v* will not be either.

在這篇論文裡，我們研究Baker-Norine的搬硬幣遊戲，並且把這個遊戲應用在離散型的熱帶因子上。特別地，我們去探討這個遊戲與等價熱帶因子之間的關係。最後我們證明了下面的定理:若$D, E$為熱帶曲線$\Gamma$上的離散型熱帶因子, 而$\overline{D}$, $\overline{E}$分別代表因子$D,E$在搬硬幣遊戲時的狀態，因子$D$與$E$等價,若且為若 $\overline{D}$可經搬硬幣遊戲變成$\overline{E}$。 / In this thesis, we study Baker-Norine's chip-firing game, and apply it to discrete tropical divisors. In particularly, we discuss the relationship between this game and the equivalence of divisors. Finally, we give a proof of the theorem: Let $D$ and $E$ be discrete tropical divisors of tropical curve $\Gamma$, and let $\overline{D}$ and $\overline{E}$ be corresponding configurations of the chip-firing game. The divisors $D$ and $E$ are equivalent if and only if $\overline{D}$ can be transformed into $\overline{E}$.

Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this set up to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as “combinatorial shadows” of curves. The development of this tropical relationship between graphs and algebraic curves has allowed for beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs. In Chapter 1 we give an overview of the history of chip-firing and its connections to algebraic geometry. In Chapter 2 we describe a reinterpretation of chip-firing in the language of partial graph orientations and apply this setup to give a new proof of the Riemann-Roch formula. We introduce and investigate transfinite chip-firing, and chip-firing with respect to open covers in Chapters 3 and 4 respectively. Chapter 5 represents joint work with Arash Asadi, where we investigate Riemann-Roch theory for directed graphs and arithmetical graphs, the latter of which are a special class of balanced vertex weighted graphs arising naturally in arithmetic geometry.

Cette thèse s'intéresse dans un premier temps aux automates cellulaires réversibles, et dans un second temps aux tas de sable linéaires. Nous construisons diverses simulations reliant les automates cellulaires aux automates à partitions, en particulier celle des automates cellulaires réversibles par les automates à partitions réversibles, ce qui était une conjecture depuis 1990. Par des constructions successives, nous montrons que le ``Billiard ball model'' de Toffoli et Margolus est capable de simuler tous les automates à partitions réversibles de dimension 2. En rassemblant ces résultats, nous montrons qu'il existe des automates cellulaires réversibles capables de simuler tous les automates cellulaires réversibles de même dimension. Dans un espace linéaire, ``Tas de sable'' et ``Chip firing game'' sont équivalents. Nous portons notre attention sur le cas où les grains tombent un à un. Des motifs délimités par des signaux apparaissent au sein des configurations engendrées. Nous étudions la dynamique du système et démontrons un équivalent asymptotique. Nous étendons nos méthodes et nos résultats à d'autres types de configurations initiales. Dans chaque cas étudié, le temps parallèle est inférieur au temps séquentiel dans un rapport de l'ordre du nombre de piles mises en œuvre.

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their Z-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.