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An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided. A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method. This argument can take advantage of structural properties of the graph: it is shown how to obtain strongerdoi:10.1007/bf01261322 fatcat:yqt5aggkuzg6ldfdrysvd63kxu