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Báo cáo toán học: "Spectral characterizations of dumbbell 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: Spectral characterizations of dumbbell graphs. | Spectral characterizations of dumbbell graphs Jianfeng Wang Department of Mathematics Qinghai Normal University Xining Qinghai 810008 P.R. China jfwang4@yahoo.com.cn Francesco Belardo Department of Mathematics University of Messina Sant Agata 98166 Messina Italy fbelardo@gmail.com Qiongxiang Huang Enzo M. Li Marzi College of Mathematics and System Science Xinjiang University Urumqi 830046 P.R. China Department of Mathematics University of Messina Sant Agata 98166 Messina Italy huangqx@xju.edu.cn emlimarzi@gmail.com Submitted Jul 13 2009 Accepted Mar 4 2010 Published Mar 15 2010 Mathematics Subject Classifications 05C50 Abstract A dumbbell graph denoted by Da b c is a bicyclic graph consisting of two vertex-disjoint cycles Ca Cb and a path Pc 3 c -1 joining them having only its end-vertices in common with the two cycles. In this paper we study the spectral characterization w.r.t. the adjacency spectrum of Da b 0 without cycles C4 with gcd a b 3 and we complete the research started in J.F. Wang et al. A note on the spectral characterization of dumbbell graphs Linear Algebra Appl. 431 2009 1707-1714 . In particular we show that Da b 0 with 3 gcd a b a or gcd a b a and b 3a is determined by the spectrum. For b 3a we determine the unique graph cospectral with Da 3a 0. Furthermore we give the spectral characterization w.r.t. the signless Laplacian spectrum of all dumbbell graphs. 1 Introduction Let G V G E G be a graph with order V G n G n and size E G m G m. Let A G be the 0 1 -adjacency matrix of G and dG v d v the degree of the vertex v. The polynomial Ộ G A det AI A G or simply G where I is the Research supported by the NSFC No. 10761008 and No. 10961023 and the XGEDU 2009 S20. Research supported by the INdAM Italy . THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R42 1 identity matrix is defined as the characteristic polynomial of G which can be written as Ộ G An ai G Ara-1 a2 G An-2 an G . Since A G is real and symmetric its eigenvalues are all real numbers. Assume