tailieunhanh - Báo cáo toán học: "Random Cayley graphs are expanders: a simple proof of the Alon–Roichman theorem"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Random Cayley graphs are expanders: a simple proof of the Alon–Roichman theorem. | Random Cayley graphs are expanders a simple proof of the Alon-Roichman theorem Zeph Landau Department of Mathematics City College of New York landau@. Alexander Russell Department of Computer Science and Engineering University of Connecticut acr@ Submitted Jul 13 2004 Accepted Aug 26 2004 Published Sep 13 2004 Mathematics Subject Classification 05C25 05C80 20C15 20F69 Abstract We give a simple proof of the Alon-Roichman theorem which asserts that the Cayley graph obtained by selecting c log G elements independently and uniformly at random from a finite group G has expected second eigenvalue no more than e here c is a constant that depends only on e. In particular such a graph is an expander with constant probability. Our new proof has three advantages over the original proof i. it is extremely simple relying only on the decomposition of the group algebra and tail bounds for operator-valued random variables ii. it shows that the log G term may be replaced with log D where D G is the sum of the dimensions of the irreducible representations of G and iii. it establishes the result above with a smaller constant c . 1 Introduction A beautiful theorem of N. Alon and Y Roichman 4 asserts that random Cayley graphs are expanders. Specifically they study the spectrum of the Cayley graphs obtained by selecting k elements independently and uniformly at random from a finite group G. They show that for every 0 there is a constant c so that the expected second eigenvalue of the normalized Supported in part by National Science Foundation grants CCR-0093065 CCR-0121277 CCR-0220264 CCR-0311368 and ElA-0218443. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R62 1 adjacency matrix of the graph is less than e as long as k ce o 1 log G . Their proof involves a clever combinatorial argument that controls the behavior of random walks taken on the random graph. Invoking established relationships between graph expansion and the second eigenvalue this implies .