  In this paper, we aim to provide a systematic study of the relationship
between Chernoff information and topological, as well as algebraic properties
of the corresponding Gaussian tree graphs for the underlying graphical testing
problems. We first show the relationship between Chernoff information and
generalized eigenvalues of the associated covariance matrices. It is then
proved that Chernoff information between two Gaussian trees sharing certain
local subtree structures can be transformed into that of two smaller trees.
Under our proposed grafting operations, bottleneck Gaussian trees, namely,
Gaussian trees connected by one such operation, can thus be simplified into two
3-node Gaussian trees, whose topologies and edge weights are subject to the
specifics of the operation. Thereafter, we provide a thorough study about how
Chernoff information changes when small differences are accumulated into bigger
ones via concatenated grafting operations. It is shown that the two Gaussian
trees connected by more than one grafting operation may not have bigger
Chernoff information than that of one grafting operation unless these grafting
operations are separate and independent. At the end, we propose an optimal
linear dimensional reduction method related to generalized eigenvalues.
