Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 591758, 10 pages doi: Research Article Ordering Unicyclic Graphs in Terms of Their Smaller Least Eigenvalues Guang-Hui Xu Department of Applied Mathematics, Zhejiang A&F University, Hangzhou 311300, China Correspondence should be addressed to Guang-Hui Xu, ghxu@ Received 15 July 2010; Accepted 2 December 2010 Academic Editor: Jozef Bana´ s ´ Copyright q 2010 Guang-Hui Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let G be a simple graph. | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 591758 10 pages doi 2010 591758 Research Article Ordering Unicyclic Graphs in Terms of Their Smaller Least Eigenvalues Guang-Hui Xu Department of Applied Mathematics Zhejiang A F University Hangzhou 311300 China Correspondence should be addressed to Guang-Hui Xu ghxu@ Received 15 July 2010 Accepted 2 December 2010 Academic Editor Jozef Banas Copyright 2010 Guang-Hui Xu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Let G be a simple graph with n vertices and let 1 G be the least eigenvalue of G. The connected graphs in which the number of edges equals the number of vertices are called unicyclic graphs. In this paper the first five unicyclic graphs on order n in terms of their smaller least eigenvalues are determined. 1. Introduction Let G be a simple graph with n vertices and let A be the 0 1 -adjacency matrix of G. We call det 1T - A the characteristic polynomial of G denoted by P G 1 or abbreviated P G . Since A is symmetric its eigenvalues 11 G 12 G . . 1n G are real and we assume that 11 G 12 G 1n G . We call 1n G the least eigenvalue of G. Up to now some good results on the least eigenvalues of simple graphs have been obtained. 1 In 1 let G be a simple graph with n vertices G Kn then 1 1n G s 1n Kn-1 . The equality holds if and only if G K1-1 where Kn-1 is the graph obtained from Kn-1 by joining a vertex of Kn-1 with K1. 2 In 2-4 let G be a simple graph with n vertices then 1nG - n 1 2 2 Journal of Inequalities and Applications The equality holds if and only if G K n 2 z n 1 2 . 3 In 5 let G be a planar graph with n 3 vertices then Xn G -G2n - 4. The equality holds if and only if G K2 n-2. 4 In 6 the author surveyed the main results of the theory of graphs with least .
