TAILIEUCHUNG - Báo cáo toán học: "On the automorphism group of integral circulant graphs"

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 the automorphism group of integral circulant graphs. | On the automorphism group of integral circulant graphs Milan Basic University of Nis Faculty of Sciences and Mathematics Visegradska 33 18000 Nis Serbia e-mail basic_milan@ Aleksandar IliC Í University of Nis Faculty of Sciences and Mathematics Visegradska 33 18000 Nis Serbia e-mail aleksandari@gmail. com Submitted Oct 6 2009 Accepted Mar 9 2011 Published Mar 31 2011 Mathematics Subject Classification 05C60 05C25 Abstract The integral circulant graph Xn D has the vertex set Zn 0 1 2 . n 1 and vertices a and b are adjacent if and only if gcd a b n E D where D d1 d2 . dk is a set of divisors of n. These graphs play an important role in modeling quantum spin networks supporting the perfect state transfer and also have applications in chemical graph theory. In this paper we deal with the automorphism group of integral circulant graphs and investigate a problem proposed in W. Klotz T. Sander Some properties of unitary Cayley graphs Electr. J. Comb. 14 2007 R45 . We determine the size and the structure of the automorphism group of the unitary Cayley graph Xn 1 and the disconnected graph Xn d . In addition based on the generalized formula for the number of common neighbors and the wreath product we completely characterize the automorphism groups Aut Xn 1 p for n being a square-free number and p a prime dividing n and Aut Xn 1 pk for n being a prime power. 1 Introduction Circulant graphs are Cayley graphs over a cyclic group. The interest of circulant graphs in graph theory and applications has grown during the last two decades. They appeared in coding theory VLSI design Ramsey theory and other areas. Recently there is vast research on the interconnection schemes based on the circulant topology - circulant graphs THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P68 1 represent an important class of interconnection networks in parallel and distributed computing see 17 . Integral circulant graphs as the circulants with integral spectra were imposed as potential .

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