Graph Spectra And Its Application In Continuous Quantum Walk

Graph spectra is an important branch of Algebraic Graph Theory, mainly con-cerns the connection of the combinatorial properties of graphs and the algebraic prop-erties of matrices. It has important application in physics, chemistry and information Theory. In recent years, the application of graph spectra in quantum computation and quantum information was well developed. Especially, in the network of quantum particles with fixed couplings. This area was intensively investigated by physicians, computer scientist and mathematicians and becomes an important topic in quantum computation.Determining which graphs are determined by their spectra is an important ques-tion in graph spectra. However, to prove whether a graph is determined by its spectra is not easy. The graphs which we have already known that are determined by their spectra are rare. Therefore it is meaningful and necessary to find more graphs which are determined by their spectra. We will study a specific class of trees which are determined by their Laplacian spectra. Moreover, we will give a characterization of their adjacent spectrum.An eigenvalue of a graph X is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Another important topic in the graph spectra is that how to describe the graphs with exactly k(k≥2) main eigenvalues. Let Xo be the graph obtained from X by deleting all pendant vertices and δ(X) the minimum degree of vertices of X. In Chapter3, we will determine all connected tricyclic graph X with δ(X0)>2and exactly two main eigenvalues.Let A be the adjacent matrix of graph X. Define a matrix value function U(t)=exp(iAt). In quantum physics, U(t) actually defines a continuous quantum walk of network with fixed couplings, which is a generalization of random walk in classic physics. We will investigate the application of graph spectra in continuous quantum walk started in Chapter4. We say we have perfect state transfer from vertex u to vertex v at time t if and only if there exits a complex number γ, such that U(t)eu=γev.It has become clear that perfect state transfer is rare. Hence, we consider a relaxation. We say we have pretty good state transfer from vertex u to vertex v if there is a complex number γ and, for each positive real number e there is a time t such that||U(t)eu-γev||<e.However, in the real experiment, we measure the elements of U(t)οU(-t). We say we have instantaneous uniform mixing, denoted by IUM for short, if U(t)οU(-t)=1/|V(X)|J. Where J is all ones matrix.We will characterize state transfer in Chapter4. First, we will study the con-nection of state transfer on quotient graphs and original graphs. And then we will investigate the property of cospectral vertices and strongly cospectral vertices. Fi-nally, we will characterize the graphs with dual degree3and perfect state transfer.In Chapter5, we will investigate the state transfer on several graph families, including double star graphs, generalized double star graphs, Caylay graphs. We will show that double star graphs don’t have perfect state transfer, but have pretty good state transfer in certain situation. Furthermore, we will give the upper bound of the pretty good state transfer in terms of time t. Secondly, we will study the state transfer on several generalized double star graphs and Cayley graphs. Following this, we will investigate alternating property of transition function by examining the transition function of path P4. Finally, we will exhibit an algebraic description of perfect state transfer and pretty good state transfer.In Chapter6, we will use Laplacian matrix substitute for adjacent matrix in the transition function and investigate which called perfect state transfer with respect to Laplacian matrix. We will characterize the state transfer with respect to Laplacian matrix. In particular, we will investigate the state transfer with Laplacian on paths and give the necessary condition for path to have pretty good state transfer with respect to Laplacian.In Chapter7, we will characterize the mixing property of continuous quantum walk on cubelike graphs, give the necessary condition for cubelike graphs to admit IUM. Furthermore, with the help of computer program, we will.give a complete list of cubelike graphs on16vertices with IUM. Finally, we will explore the connection of perfect state transfer and IUM on graphs and their complements.