Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The spectral radius and the maximum degree of irregular graphs. | The spectral radius and the maximum degree of irregular graphs Sebastian M. Cioaba Department of Mathematics University of California San Diego La Jolla CA 92093-0112 scioaba@ Submitted Jan 20 2007 Accepted May 3 2007 Published May 23 2007 MR Subject Classifications 05C50 15A18 Abstract Let G be an irregular graph on n vertices with maximum degree A and diameter D. We show that A - Al nD where Al is the largest eigenvalue of the adjacency matrix of G. We also study the effect of adding or removing few edges on the spectral radius of a regular graph. 1 Preliminaries Our graph notation is standard see West 22 . For a graph G we denote by Aj G the i-th largest eigenvalue of its adjacency matrix and we call A1 G the spectral radius of G. If G is connected then the positive eigenvector of norm 1 corresponding to A1 G is called the principal eigenvector of G. The spectral radius of a connected graph has been well studied. Results in the literature connect it with the chromatic number the independence number and the clique number of a connected graph 9 11 12 17 23 . Recently it has been shown that the spectral radius also plays an important role in modeling virus propagation in networks 1o 21 . In this paper we are interested in the connection between the spectral radius and the maximum degree A of a connected graph G. In particular we study the spectral radius of graphs obtained from A-regular graphs on n vertices by deleting a small number of edges or loops. The Erdos-Renyi graph ER q is an example of such a graph see 9 15 and the references within for more details on its spectral radius and other interesting properties. Research partially supported by an NSERC postdoctoral fellowship. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R38 1 It is a well known fact that Al G A G with equality if and only if G is regular. It is natural to ask how small A G A1 G can be when G is irregular. Cioaba Gregory and Nikiforov 5 proved that if G is an irregular graph on
