TAILIEUCHUNG - Báo cáo toán học: "Integral Cayley graphs over abelian groups"

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: Integral Cayley graphs over abelian groups. | Integral Cayley graphs over abelian groups Walter Klotz and Torsten Sander Institut fur Mathematik Technische Universitat Clausthal Germany klotz@ Submitted Dec 8 2009 Accepted May 20 2010 Published May 25 2010 Mathematics Subject Classification 05C25 05C50 Abstract Let r be a finite additive group S c r 0 E S S s s E S S. The undirected Cayley graph Cay r S has vertex set r and edge set a b a b E r a b E S . A graph is called integral if all of its eigenvalues are integers. For an abelian group r we show that Cay r S is integral if S belongs to the Boolean algebra B r generated by the subgroups of r. The converse is proven for cyclic groups. A finite group r is called Cayley integral if every undirected Cayley graph over r is integral. We determine all abelian Cayley integral groups. 1 Introduction Eigenvalues of an undirected graph G are the eigenvalues of an arbitrary adjacency matrix of G. Harary and Schwenk 9 defined G to be integral if all of its eigenvalues are integers. Since then many integral graphs have been discovered for a survey see 4 . Nevertheless as is shown in 2 the probability of a labeled graph on n vertices to be integral is at most 2-n 400 for sufficiently large n. Known characterizations of integral graphs are restricted to certain graph classes. Here we proceed towards a characterization of integral Cayley graphs over abelian groups. Let r be a finite additive group S c r 0 E S S s s E S S. The undirected Cayley graph Cay r S has vertex set r. Vertices a b E r are adjacent if a b E S. For general properties of Cayley graphs we refer to Godsil and Royle 8 or Biggs 5 . Abdollahi and Vatandoost 1 show that there are exactly seven connected cubic integral Cayley graphs. So 15 presents a characterization of integral circulant graphs which are Cayley graphs over cyclic groups. In this paper we prove for an abelian group r that Cay r S is integral if S belongs to the Boolean algebra B r .

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