TAILIEUCHUNG - Báo cáo toán học: "Integral Quartic Cayley Graphs on 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 Quartic Cayley Graphs on Abelian Groups. | Integral Quartic Cayley Graphs on Abelian Groups A. Abdollahi Department of Mathematics University of Isfahan Isfahan 81746-73441 Iran and School of Mathematics Institute for Research in Fundamental Sciences IPM 19395-5746 Tehran Iran. E. Vatandoost Department of Mathematics University of Isfahan Isfahan 81746-73441 Iran Submitted Aug 12 2010 Accepted Mar 29 2011 Published Apr 14 2011 Mathematics Subject Classifications 05C25 05C50 Abstract A graph is called integral if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly 27 connected integral Cayley graphs up to 11 vertices. 1 Introduction and Results A graph is called integral if all the eigenvalues of its adjacency matrix are integers. The notion of integral graphs was first introduced by Harary and Schwenk in 1974 13 . It is known that the number of non-isomorphic k-regular integral graphs is finite See . 10 . Bussemaker and Cvetkovic 8 and independently Schwenk 20 proved that there are exactly 13 connected cubic integral graphs. It is shown in 4 and 5 that there are exactly 263 connected integral graphs on up to 11 vertices. Radosavljevic and Simic in 19 determined all thirteen nonregular nonbipartite connected integral graphs with maximum degree four. Stevanovic 22 determined all connected 4-regular integral graphs avoiding 3 in the spectrum. A survey of results on integral graphs may be found in 6 . Omidi 17 identified integral graphs with at most two cycles with no eigenvalues 0. Sander 18 proved that Sudoku graphs are integral. In 2 it is shown that the total number of adjacency matrices of integral graphs with n vertices is less than or equal to 2400 for a sufficiently large n. Let G be a non-trivial group with the identity element 1 and let S be a non-empty subset of G 1 such that S S-1 s-1 s G S . The THE ELECTRONIC JOURNAL

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