tailieunhanh - Báo cáo toán học: "The diameter and Laplacian eigenvalues of directed graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The diameter and Laplacian eigenvalues of directed graphs. | The diameter and Laplacian eigenvalues of directed graphs Fan Chung University of California San Diego La Jolla CA 92093-0112 Submitted Feb 13 2006 Accepted Feb 14 2006 Published Feb 22 2006 Mathematics Subject Classification 05C20 Abstract For undirected graphs it has been known for some time that one can bound the diameter using the eigenvalues. In this note we give a similar result for the diameter of strongly connected directed graphs G namely D G 2 min. log 1 3U log 1 where A is the first non-trivial eigenvalue of the Laplacian and 3 is the Perron vector of the transition probability matrix of a random walk on G. 1 Introduction For an undirected graph G on n vertices we can upper bound the diameter D G by using eigenvalues of the Laplacian 1 2 as follows. D G log n 1 log An 1 A1 An 1 - A1 J 1 where 0 Ao Al An-1 denote the eigenvalues of the Laplacian of G. A natural question is to see if it is feasible to extend such relations to directed graphs. In a directed graph the distance and diameter can be naturally defined The distance from a vertex u to a vertex v is the length of a shortest directed path from u to v. A directed graph is strongly connected if for any two vertices u and v there is a directed path from u to v . The diameter of a strongly connected directed graph is defined to be the maximum distance among pairs of vertices. If a directed graph is not strongly connected its diameter is taken to be infinity. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N4 1 In a recent paper 4 the Laplacian of a directed graph was dehned. The eigenvalues of this Laplacian turn out to be quite useful for capturing various isoperimetric properties of directed graphs. For example the spectral gap of the Laplacian can be used to bound the mixing rate of random walks in directed graphs. The eigenvalues of the Laplacian were also used to establish a general Cheeger inequality. In this note we will establish the following diameter bound by using the spectral gap of the .