Báo cáo toán học: "A Note on Commuting Graphs for Symmetric Groups"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Note on Commuting Graphs for Symmetric Groups. | A Note on Commuting Graphs for Symmetric Groups C. Bates 30 Holtdale Lawn Holt Park Leeds LS16 7RP United Kingdom chrisjbates@ S. Hart School of Economics Mathematics Statistics Birkbeck College Malet Street London WC1E 7HX United Kingdom D. Bundy Adalbertstrasse 17 24106 Kiel Germany davidbundy@ P. Rowley School of Mathematics University of Manchester PO Box 88 Manchester M60 1QD United Kingdom Submitted May 13 2008 Accepted Dec 15 2008 Published Jan 7 2009 Mathematics Subject Classification 05C25 Abstract The commuting graph C G X where G is a group and X a subset of G has X as its vertex set with two distinct elements of X joined by an edge when they commute in G. Here the diameter and disc structure of C G X is investigated when G is the symmetric group and X a conjugacy class of G. 1 Introduction The purpose of this note is to record certain properties of commuting graphs C G X where G is Sym n the symmetric group of degree n and X is a G-conjugacy class. In 1 C G X was investigated when X was a conjugacy class of involutions. There it was shown that C G X is connected unless n 2t 1 or n 4 1 being the number of 2-cycles in the involution and that the diameter of C G X is at most 3 except for three specifically given graphs when n 2 6 8 10g . Moreover if we exclude these three exceptional graphs an algorithm is given which determines the distance between two vertices using data encoded in x-graphs see Lemma and Proposition in 1 for more details . For further recent work on commuting graphs we direct the reader to 2 3 and 4 . THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R6 1 We recall that C G X is the graph whose vertex set is X with X y 2 X x y joined whenever they commute. Clearly elements of G induce by conjugation graph automorphisms of C G X . Since G is transitive on the vertices of C G X we may without loss choose a to be a fixed element of X. Writing a as a product of .