In this paper, we want to find out the determining factors of Chernoff information in distinguishing a set of Gaussian graphs. We find that Chernoff information of two Gaussian graphs can be determined by the generalized eigenvalues of their covariance matrices. We find that the unit generalized eigenvalue doesn't affect Chernoff information and its corresponding dimension doesn't provide information for classification purpose. In addition, we can provide a partial ordering using Chernoff information between a series of Gaussian trees connected by independent grafting operations. With the relationship between generalized eigenvalues and Chernoff information, we can do optimal linear dimension reduction with least loss of information for classification.