tailieunhanh - Báo cáo toán học: "The Laplacian Spread of Tricyclic Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The Laplacian Spread of Tricyclic Graphs. | The Laplacian Spread of Tricyclic Graphs Yanqing Chen 1 and Ligong Wang2 i Department of Applied Mathematics Northwestern Polytechnical University Xi an Shaanxi 710072 P. R. China. 1yanqing_chen@ 2ligongwangnpu@ Submitted Nov 28 2008 Accepted Jun 23 2009 Published Jul 2 2009 Mathematics Subject Classifications 05C50 15A18. Abstract The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper we investigate Laplacian spread of graphs and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order. 1 Introduction In this paper we consider only simple undirected graphs. Let G V E be a graph with vertex set V V G v1 v2 . vn and edge set E E G . The adjacency matrix of the graph G is defined to be a matrix A A G aij of order n where aij 1 if vi is adjacent to Vj and aij 0 otherwise. The spectrum of G can be denoted by S G A1 G A2 G . An G where A1 G A2 G An G are the eigenvalues of A G arranged in weakly decreasing order. The spread of graph G is defined as SA G A1 G An G . Generally the spread of a square complex matrix M is defined to be s M ma. . Ai Aj where the maximum is taken over all pairs of eigenvalues of M. There have been some studies on the spread of an arbitrary matrix 8 15 17 18 . Recently the spread of a graph has received much attention. In 16 Petrovic determines all minimal graphs whose spread do not exceed 4. In 6 Gregory Hershkowitz and Supported by the National Natural Science Foundation of China the Natural Science Basic Research Plan in Shaanxi Province of China and SRF for ROCS SEM. 1 Corresponding author. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R80 1 Kirkland present some lower and upper bounds for the spread of a graph. They show that the path is the unique graph with minimum spread among connected graphs