tailieunhanh - Báo cáo toán học: Periodic Graphs

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Periodic Graphs. | Periodic Graphs Chris Godsil Combinatorics Optimization University of Waterloo cgodsil@ Submitted Nov 4 2010 Accepted Jan 17 2011 Published Jan 26 2011 Mathematics Subject Classification 05C50 81P68 Abstract Let X be a graph on n vertices with adjacency matrix A and let H t denote the matrix-valued function exp iAt . If u and v are distinct vertices in X we say perfect state transfer from u to v occurs if there is a time T such that H t u v 1. If u G V X and there is a time Ơ such that H ơ u u 1 we say X is periodic at u with period Ơ. It is not difficult to show that if the ratio of distinct nonzero eigenvalues of X is always rational then X is periodic. We show that the converse holds from which it follows that a regular graph is periodic if and only if its eigenvalues are distinct. For a class of graphs X including all vertex-transitive graphs we prove that if perfect state transfer occurs at time T then H t is a scalar multiple of a permutation matrix of order two with no fixed points. Using certain Hadamard matrices we construct a new infinite family of graphs on which perfect state transfer occurs. 1 Introduction Let X be a graph with adjacency matrix A. We define the matrix function H t by t n H t exp itA inAn ị. We note that t is a real variable H t exp -iAt H t -1 and therefore H t is a unitary matrix. We have H 0 I and H s t H s H t . THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P23 1 We say that graph is periodic with respect to the vector z if there is a real number T such that H t z is a scalar multiple of z. Since H t is unitary this scalar will have absolute value 1. We say that X is periodic relative to the vertex u if it is periodic relative to the standard basis vector eu or equivalently if there is a time T such that H t u u 1. We say that X itself is periodic if there is a time T such that H t is diagonal. If there are basis vectors z1 . zn such that X is periodic with period Tr relative to the vector zr for r 1 . n then it .

• Công nghệ - Môi trường
• báo cáo của tạp chí Department of Mathematic
• tài liệu báo cáo nghiên cứu khoa học
• cách trình bày báo cáo
• kiến thức toán học
• báo cáo toán học