A Quantum Probability Approach To Perfect State Transfers In Graphs

Markov chains or random walks on graphs have proved to be a valid tool in many research fields. As quantum analogues of random walks, quantum walks on graphs have attracted much interest from both mathematicians and computer scientists in recent years. A quantum walk on a graph is naturally defined as a unitary transformation on the tensor product of the Hilbert space of the graph and the auxiliary Hilbert space, and with the property that the probability amplitude is non zero only on edges of the graph. In this dissertation, we mainly apply the quantum walk approach to a topic on perfect state transfers in graphs. We take into account the representation of a continuous-time quantum walk in a graph X by the matrix e-itA(X).We provide criteria for checking whether or not there are perfect state transfers in a graph. Using these results, we provide several new examples of perfect state transfer in simple graphs. Furthermore, by using the method of spectral decomposition, we examine joins, direct products and mixings of graphs with emphasizing perfect state transfers in them.This dissertation is organized as follows.In Chapter 1, we briefly describe the research history and development status of graph theory, and give some fundamental notions such as distance-regular graphs, association schemes, perfect state transfer, characterization and Hadamard matrices.Chapter 2 focuses on a topic on perfect state transfers in graphs. Indeed, we apply the quantum probability approach to graphs such as distance-regular graphs, graphs belonging to association schemes and Cubelike graphs. Several results are obtained on perfect state transfers in these graphs.Finally in Chapter 3, by using the method of spectral decomposition, we examine joins, direct products and mixings of graphs with emphasizing perfect state transfers on them.