TAILIEUCHUNG - Báo cáo toán học: "The orderings of bicyclic graphs and connected graphs by algebraic connectivity"

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: The orderings of bicyclic graphs and connected graphs by algebraic connectivity. | The orderings of bicyclic graphs and connected graphs by algebraic connectivity Jianxi Li Department of Mathematics Information Science Zhangzhou Normal University Zhangzhou Fujian P. R. China fzjxli@ Ji-Ming Guo Department of Applied Mathematics China University of Petroleum Dongying Shandong P. R. China j imingguo@ Wai Chee Shiu Department of Mathematics Hong Kong Baptist University Kowloon Tong Hong Kong P. R. China wcshiu@ Submitted May 31 2010 Accepted Nov 15 2010 Published Dec 3 2010 Mathematics Subject Classifications 05C50 Keywords bicyclic graph connected graph algebraic connectivity order Abstract The algebraic connectivity of a graph G is the second smallest eigenvalue of its Laplacian matrix. Let Bn be the set of all bicyclic graphs of order n. In this paper we determine the last four bicyclic graphs according to their smallest algebraic connectivities among all graphs in Bn when n 13. This result together with our previous results on trees and unicyclic graphs can be used to further determine the last sixteen graphs among all connected graphs of order n. This extends the results of Shao et al. The ordering of trees and connected graphs by their algebraic connectivity Linear Algebra Appl. 428 2008 1421-1438 . Supported by the National Science Foundation of China the Fundamental Research Funds for the Central Universities FRG Hong Kong Baptist University. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R162 1 1 Introduction Let G be a simple graph with vertex set V G v1 v2 . vn and edge set E G . For v G V G let NG v or N v for short be the set of vertices which are adjacent to v in G and d v N v be the degree of v. For any e G E G we use G e to denote the graph obtained by deleting e from G. Readers are referred to 2 for undefined terms. Let A G and D G be the adjacency matrix and the diagonal matrix of vertex degrees of G respectively. The Laplacian matrix of G is defined as L G D G A G . It is .

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