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 @r(G) be the spectral radius of G and @l(G) be the smallest eigenvalue of G. The join [email protected]?H 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"5minor-free graph, [email protected](G)= =5 vertices, [email protected](G)>=-3n-9,where equality holds if and only if G is isomorphic to K"3","n"-"3.