Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: New regular partial difference sets and strongly regular graphs with parameters (96,20,4,4) and (96,19,2,4). | New regular partial difference sets and strongly regular graphs with parameters 96 20 4 4 and 96 19 2 4 Anka Golemac Josko Mandic and Tanja Vucicic University of Split Department of Mathematics Teslina 12 III 21000 Split Croatia golemac@ majo@ vucicic@ Submitted Sep 15 2005 Accepted Sep 29 2006 Published Oct 19 2006 Mathematics Subject Classification 05B05 05B10 05E30 Abstract New 96 20 4 4 and 96 19 2 4 regular partial difference sets are constructed together with the corresponding strongly regular graphs. Our source are 96 20 4 regular symmetric designs. Keywords Difference set partial difference set Cayley graph symmetric design. 1 Introduction and preliminaries We start with defining objects to be constructed. Definition 1 Let H be a group of order v. A k-subset S c H is called a V k X p partial dif ference set if the multiset xy-1 I X y 2 S X y contains each nonidentity element of S exactly X times and it contains each nonidentity element of H n S exactly p times. Using the notation of a group ring ZH where S 12s2S s a V k X p partial difference set S c H in the group H can be defined as a subset for which the equation S S -1 k e XS n e p H n S n e holds e denotes the group identity element. Partial differential sets S1 and S2 in groups H1 and H2 respectively we will call equivalent if there exists a group isomorphism H1 H2 which maps S1 onto S2. The notion of a partial difference set generalizes that of a difference set well-known in group and design theory. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R88 1 Definition 2 A v k A difference set is a k-element subset A c H in a group H of order v provided that the multiset xy-1 I x y 2 A x yg contains each nonidentity element of H exactly A times. In terms of a group ring A c H is a difference set in a group H if and only if the relation A A 1 k eg AH n eg holds in ZH. In case a set A c H is a difference set in a group H its so called shift Ax by each element x 2 H is a difference
