The first (M-1) and second (M-2) Zagreb indices and the spectral radius (rho) are topological indices. This Letter presents upper bounds for these indices of triangle- and quadrangle-free connected graphs, in terms of the number of vertices (n), number of edges (m), and the radius (r). These bounds are: M-1(G) <= n(n + 1 - r), M-2(G) <= m(n + 1 - r), rho <= root n + 1 -r, and we determine the graphs for which the bounds are attained. (c) 2008 Elsevier B. V. All rights reserved.