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Báo cáo toán học: "On restricted unitary Cayley graphs and symplectic transformations modulo n"

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: On restricted unitary Cayley graphs and symplectic transformations modulo n. | On restricted unitary Cayley graphs and symplectic transformations modulo n Niel de Beaudrap Quantum Information Theory Group Institut fur Physik und Astronomie Universitat Potsdam Submitted Feb 12 2010 Accepted Apr 27 2010 Published May 7 2010 Mathematics Subject Classification 05C12 05C17 05C50 Abstract We present some observations on a restricted variant of unitary Cayley graphs modulo n and implications for a decomposition of elements of symplectic operators over the integers modulo n. We define quadratic unitary Cayley graphs Gn whose vertex set is the ring Zn and where residues a b modulo n are adjacent if and only if their difference is a quadratic residue. By bounding the diameter of such graphs we show an upper bound on the number of elementary operations symplectic scalar multiplications symplectic row swaps and row additions or subtractions required to decompose a symplectic matrix over Zn. We also characterize the conditions on n for Gn to be a perfect graph. 1 Introduction For an integer n 1 we denote the ring of integers modulo n by Zn and the group of multiplicative units modulo n by Z . A well-studied family of graphs are the unitary Cayley graphs on Zn which are defined by Xn Cay Zn Z . These form the basis of the subject of graph representations 1 and are also studied as objects of independent interest see for example 2-5 . We consider- a subgraph Gn Xn of the unitary Cayley graphs defined as follows. Let Qn u2 I u E Z be the group of quadratic units modulo n quadratic residues which are also multiplicative units and Tn Qn. We then define Gn Cay Zn Tn in which two vertices given by a b E Zn are adjacent if and only if their difference is a quadratic unit in Zn i.e. if a b E u2 I u E Z . In the case where n 1 mod 4 and is prime Gn coincides with the Paley graph on n vertices thus the graphs Gn are a niel. debeaudrap@gmail. com THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R69 1 circulant generalization of these graphs for arbitrary n. We refer to