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Báo cáo toán hoc:"On the unitary Cayley graph of a finite ring"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On the unitary Cayley graph of a finite ring. | On the unitary Cayley graph of a finite ring Reza Akhtar Department of Mathematics Miami University reza@calico.mth.muohio.edu Tiffany Jackson-Henderson St. Augustine s College tj ackhend @yahoo.com Rachel Karpman Scripps College rkarpman@scrippscollege.edu Megan Boggess Columbia Union College meganboggess@gmail.com Isidora Jimenez Mills College ijimenez@mills.edu Amanda Kinzel Department of Mathematics Purdue University asphilli@math.purdue.edu Dan Pritikin Department of Mathematics Miami University pritikd@muohio.edu Submitted May 16 2009 Accepted Sep 7 2009 Published Sep 18 2009 Mathematics Subject Classification 05C25 05C30 Abstract We study the unitary Cayley graph associated to an arbitrary finite ring determining precisely its diameter girth eigenvalues vertex and edge connectivity and vertex and edge chromatic number. We also compute its automorphism group settling a question of Klotz and Sander. In addition we classify all planar graphs and perfect graphs within this class. 1 Introduction Given an integer n consider the graph Cay Zn zn with vertex set Zn the integers modulo n with vertices x and y adjacent exactly when x y is a unit in the ring Zn. These so-called unitary Cayley graphs have been studied as objects of independent interest see for example 2 3 7 8 9 but are of particular relevance in the study of graph THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R117 1 representations begun in 5 and continued in many other papers. A graph is said to be representable modulo n if it is isomorphic to an induced subgraph of Cay Zn zn the central problem in graph representations is to determine the smallest positive n modulo which a given graph G is representable. It is natural then to study unitary Cayley graphs in the hope of gaining insight into the graph representation problem. A generalization of unitary Cayley graphs presents itself readily given a finite ring R commutative with unit element 1 0 one may define GR Cay R R to be the ring whose vertex set .