Variances and determinantal profiles of orientations

Given a simple graph G, let X be the random variable which is the determinant of the (oriented) adjacency matrix of an orientation of G. It is known that the expectation E(X) equals the number of perfect matchings of G. In this paper we give a graphical interpretation of the variance Var(X). We also give complete determinantal profiles of several classes of graphs, including wheels, fans, and general books.