tailieunhanh - Báo cáo hóa học: " Research Article Bounds of Eigenvalues of K3,3-Minor Free Graphs"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Bounds of Eigenvalues of K3,3-Minor Free Graphs | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 852406 5 pages doi 2009 852406 Research Article Bounds of Eigenvalues of K3 3-Minor Free Graphs Kun-Fu Fang Faculty of Science Huzhou Teachers College Huzhou 313000 China Correspondence should be addressed to Kun-Fu Fang kffang@ Received 17 February 2009 Accepted 11 May 2009 Recommended by Wing-Sum Cheung The spectral radius p G of a graph G is the largest eigenvalue of its adjacency matrix. Let 1 G be the smallest eigenvalue of G. In this paper we have described the K3 3-minor free graphs and showed that A let G be a simple graph with order n 7. If G has no K3 3-minor then p G 1 3n - 8. B Let G be a simple connected graph with order n 3. If G has no K3 3-minor then 1 G - 2n - 4 where equality holds if and only if G is isomorphic to K2n-2. Copyright 2009 Kun-Fu Fang. 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. 1. Introduction In this paper all graphs are finite undirected graphs without loops and multiple edges. Let G be a graph with n n G vertices m m G edges and minimum degree ỗ or ỗ G . The spectral radius p G of G is the largest eigenvalue of its adjacency matrix. Let 1 G be the smallest eigenvalue of G. The join GVH is the graph obtained from G u H by joining each vertex of G to each vertex of H. A graph H is said to be a minor of G if H can be obtained from G by deleting edges contracting edges and deleting isolated vertices. A graph G is H-minor free if G has no H -minor. Brualdi and Hoffman 1 showed that the spectral radius satisfies p G k - 1 where m k k - 1 2 with equality if and only if G is isomorphic to the disjoint union of the complete graph Kk and isolated vertices. Stanley 2 improved the above result. Hong et al. 3 showed that if G is a simple connected graph then p Ỗ - 1 ự