Bounds on the error in the normal approximation of sums of dependent random variables introduced by Stein are interpreted in terms of dependency graphs. This leads to improvements on a central limit theorem of Petrovskaya and Leontovich and recent applications by Baldi and Rinott. In particular, bounds on rates of convergence are obtained. As an application we study the normal approximation to the number of local maxima of a random function on a graph.
Pierre Baldi. Yosef Rinott. "On Normal Approximations of Distributions in Terms of Dependency Graphs." Ann. Probab. 17 (4) 1646 - 1650, October, 1989. https://doi.org/10.1214/aop/1176991178