On the study of the Potts model, Tutte and Chromatic Polynomials ...

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Contents I. Introduction to Graph Theory 3 A. Graphs and Subgraphs 3 B. Graph Coloring 4 C. The Chromatic Polynomial 4 D. Deletion-Contraction Property 6 E. The Tutte Polynomial 7 F. Chromatic Roots 7 II. Introduction to the Potts Model 10 A. A Model for Interacting Spins 10 B. The Potts Model Partition Function 11 C. The Potts Model Partition Function and the Tutte Polynomial 12 D. The Antiferromagnetic Potts Model 14 E. Exact Calculations on Recursive Graphs for the Potts Partition Function 15 F. Thermodynamics Functions and Phase Transition 16 III. Some Connections to Quantum Computing 18 A. Partition Function Algorithm (Superpolynomial Speed) 18 B. Knot Invariants Algorithms (Superpolynomial Speed) 19 C. Applications on Topological Quantum Computing 20 IV. Discussion 22 References 24 2
I. INTRODUCTION TO GRAPH THEORY A. Graphs and Subgraphs In mathematics and physics, there are several problems where one is interested in calculating a function on a graph, G = G(V, E), where V is the set of vertices (sites) and E is the set of edges (bonds) [1]. FIG. 1: (Right) Petersen Graph is an example of a 3-regular graph, with 10 vertices, 15 edges, and where each vertex has 3 neighbors. (Left) It is also an example of a non-planar graph, i.e., there are crossing edges [2]. Definition I.1 A spanning subgraph G ′ of G is G ′ = (V, E ′ ) with E ′ ⊆ E, i.e., it has the same vertices and a subset of the edges of G [1]. Definition I.2 A connected graph with no cycles is a tree. Definition I.3 A spanning subgraph G ′ with no cycles is a spanning forest of G. Definition I.4 The complete graph K n is the( graph with n vertices such that each pair of n vertices is connected by an edge, so E(K n ) = . 2) Definition I.5 A loop is an edge that connects a vertex to itself. P (G, q) vanishes if G contains one or more loops. Definition I.6 A recursive family of graphs G m is a family for which the (m+1)’th member is obtained from the m’th member graph either by gluing on some fixed subgraph or by cutting through the m’th member, inserting the subgraph and gluing the graph together. 3
Page 1: On the study of the Potts model, Tu
Page 5 and 6: FIG. 2: In 1852, Guthrie noticed th
Page 7 and 8: FIG. 5: The deletion and contractio
Page 9 and 10: G. Algorithm to Chromatic Roots Com
Page 11 and 12: adjacent spins sharing common state
Page 13 and 14: to write the Tutte polynomial of a
Page 15 and 16: FIG. 7: The q-state Potts model mod
Page 17 and 18: The ground state degeneracy (per si
Page 19 and 20: B. Knot Invariants Algorithms (Supe
Page 21 and 22: uilding them. A fundamental challen
Page 23 and 24: Another current related field of st
Page 25 and 26: [24] Van den Nest et al, 2008, Quan

On the study of the Potts model, Tutte and Chromatic Polynomials ...

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