It is proved that the smallest eigenvalue of (the adjacency matrix of) a non-bipartite graph on \(n\) vertices is at least \(-\Delta+ 1/(D+ 1)n\) where \(\Delta\) is the maximum degree and \(D\) the diameter. In addition, the precise approximation guarantee of the max cut algorithm of M. X. Goemans and D. P. Williamson [J. Assoc. Comput. Mach. 42, No. 6, 1115-1145 (1995; Zbl 0885.68088)] is determined for some classes of graphs.