Tree-width, clique-minors, and eigenvalues

Let G be a simple graph with n vertices and tw(G) be the tree-width of G. Let (G) be the spectral radius of G and lambda(G) be the smallest eigenvalue of G. The join GdelH of disjoint graphs of G and H is the graph obtained from G + H by joining each vertex of G to each vertex of H. In this paper, several results which are concerned with tree-width, clique-minors, and eigenvalues of graphs are given. In particular, we have (1) If G is K-5 minor-free graph, then rho(G) less than or equal to 1 + root3n-8, where equality holds if and only if G is isomorphic to K(3)del(n-3)K-1. (2) If G is K-5 minor-free graph with n greater than or equal to 5 vertices, then lambda(G) greater than or equal to - root3n-9, where equality holds if and only if G is isomorphic to K-3,K-n-3 (C) 2003 Elsevier B.V. All rights reserved.