tailieunhanh - Báo cáo toán học: "Recognizing circulant graphs in polynomial time: An application of association schemes"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Recognizing circulant graphs in polynomial time: An application of association schemes. | Recognizing circulant graphs in polynomial time An application of association schemes Mikhail E. Muzychuk Department of Computer Science and Mathematics Netanya Academic College Netanya 42365 Israel muzy@ Gottfried Tinhofer Zentrum Mathematik Technical University of Munich 80290 Munich Germany gottin@ Submitted October 7 2000 Accepted May 26 2001 MR subject classifications 05C25 05C85. Abstract In this paper we present a time-polynomial recognition algorithm for certain classes of circulant graphs. Our approach uses coherent configurations and Schur rings generated by circulant graphs for elucidating their symmetry properties and eventually finding a cyclic automorphism. Key words Circulant graphs association schemes Schur rings. 1 Introduction We consider graphs of the form G X y where X is a finite set and Y is a binary relation on X the adjacency relation. For x 2 X put y x fy X y 2 Y Partially supported by DAAD fellowship A 00 24054 THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R26 1 Let G be a group and G X y a graph with vertex set X G and with adjacency relation Y dehned with the aid of some subset S c G by Y 9 h g h 2 G A hg 1 2 S . Then G is called Cayley graph over the group G and S is called connection set of G. Let Zn n 2 N stand for a cyclic group of order n written additively. A circulant graph G of order n or a circulant for short is a Cayley graph over Zn. In this particular case the adjacency relation Y has the form n 1 Y u w X i Y 0 i 0 where Y 0 is the set of successors of the vertex 0. Evidently the set of successors y ì of an arbitrary vertex i satishes Y i i Y 0 . All arithmetic operations with vertex numbers are understood modulo n. We do not distinguish by notation between the element z 2 Zn and the integer z 2 Z. From the context it will always be clear what is meant. For a 2 Zn and S c Zn we write aS for the set as I s 2 S . For a circulant G the connection set is y 0 . G is a simple undirected graph