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 . China jfwang4@ Francesco Belardo Department of Mathematics University of Messina Sant Agata 98166 Messina Italy fbelardo@ Qiongxiang Huang Enzo M. Li Marzi College of Mathematics and System Science Xinjiang University Urumqi 830046 . China Department of Mathematics University of Messina Sant Agata 98166 Messina Italy huangqx@ emlimarzi@ 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 . the adjacency spectrum of Da b 0 without cycles C4 with gcd a b 3 and we complete the research started in . 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 . 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
