Asymptotic normality of some graph-related statistics

Petrovskaya and Leontovich proved a central limit theorem for sums of dependent random variables indexed by a graph. The authors apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. They briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.