Applications of Combinatorial Analysis to the Calculation of the Partition Function of the Ising Model

Abstract

The research work discussed in this thesis investigated the application of combinatorics and graph theory in the analysis of the partition function of the Ising Model. Chapter 1 gives a general introduction to the partition function of the Ising Model and the Feynman Identity in the language of graph theory. Chapter 2 describes and proves combinatorially the Feynman Identity in the special case when there is only one vertex and multiple loops. Chapter 3 digresses into the number of cycles in a directed graph, along with its application in the special case to derive the analytical expression of the number of non-periodic cycles with positive and negative signs. Chapter 4 comes back to the general case of the Feynman Identity. The Feynman Identity is applied to several special cases of the graph and a combinatorial identity is established for each case. Chapter 5 concludes the thesis by summarizing the main ideas in each chapter.