TAILIEUCHUNG - Báo cáo toán học: "Improved Upper Bounds for the Laplacian Spectral Radius of a Graph"

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: Improved Upper Bounds for the Laplacian Spectral Radius of a Graph. | Improved Upper Bounds for the Laplacian Spectral Radius of a Graph Tianfei Wang1 1 School of Mathematics and Information Science Leshan Normal University Leshan 614004 . China 1wangtf818@ Jin Yang2 Bin Li3 2 3Laboratory of Intelligent Information Processing and Application Leshan Normal University Leshan 614004 . China 2jinnyang@ 3zishan818@ Submitted May 16 2010 Accepted Jan 24 2011 Published Feb 14 2011 Mathematics Subject Classifications O5C50 15A18 Abstract In this paper we present three improved upper bounds for the Laplacian spectral radius of graphs. Moreover we determine all extremal graphs which achieve these upper bounds. Finally some examples illustrate that the results are best in all known upper bounds in some sense. 1 Introduction Let G be a simple connected graph with vertex set V G v1 v2 vn and edge set E G e1 e2 em . For any vertex v E V G the degree of v the set of neighbors of v and the average of the degrees of the vertices adjacent to v are denoted by dv Nv and mv respectively. Let D G diag du u E V G and A G auv be the diagonal matrix of the vertex degrees and the 0 1 adjacency matrix of G respectively. Then the matrix L G D G A G is called the Laplacian matrix of a graph G. Obviously it is symmetric and positive semi-definite and consequently its eigenvalues are nonnegative real numbers. In addition since each row sum of L G is 0 0 is the smallest eigenvalue of L G . Therefore the eigenvalues of L G which are called the Laplacian eigenvalues of G can be denoted by A1 G A2 G An G 0 where A1 G A G is also called the Laplacian spectral radius. Let K G D G A G . It is called the signless Laplacian matrix of G or Quasi-Laplacian . A semiregular bipartite graph G V E is a graph with bipartition V1 V2 of V such that all vertices in V have the same degree kị THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P35 1 for i 1 2. The Laplacian eigenvalues of a graph are important in graph theory because they have close relations .

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