Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Revisiting two classical results on graph spectra. | Revisiting two classical results on graph spectra Vladimir Nikiforov Department of Mathematical Sciences University of Memphis Memphis TN 38152 USA vnikifrv@ Submitted Sep 4 2006 Accepted Dec 18 2006 Published Jan 17 2007 Mathematics Subject Classifications 05C50 Abstract Let p G and ímin G be the largest and smallest eigenvalues of the adjacency matrix of a graph G. Our main results are i If H is a proper subgraph of a connected graph G of order n and diameter p G n ii If G is a connected nonbipartite graph of order n and diameter D then 2 h G fimin G G 2D For large p and D these bounds are close to the best possible ones. Keywords smallest eigenvalue largest eigenvalue diameter connected graph bipartite graph 1 Introduction Our notation is standard . see 2 3 and 5 . In particular unless specified otherwise all graphs are defined on the vertex set n f 1 . ng and ụ G and ụmin G stand for the largest and smallest eigenvalues of the adjacency matrix of a graph G. The aim of this note is to refine quantitatively two well-known results on graph spectra. The first one following from Frobenius s theorem on nonnegative matrices asserts that if H is a proper subgraph of a connected graph G then ụ G ụ H . The second one due to H. Sachs 7 asserts that if G is a connected nonbipartite graph then ụ G ụmin G . Our main result is the following theorem. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R14 1 Theorem 1 If H is a proper subgraph of a connected graph G of order n and diameter D then M G - M H M y. n. 1 It can be shown that for large M and D the right-hand of 1 gives the correct order of magnitude examples can be constructed as in the proofs of Theorems 2 and 3. Theorem 2 If G is a connected nonbipartite graph of order n and diameter D then M G I Mmin G 2. M G n 2 Moreover for all k 3 D 4 and n D 2k 1 there exists a connected nonbipartite graph G of order n and diameter D with M G k and M G Mmin G 4. k - 1 2D_1 Theorem 2 shows that M G Mmin G can be .
