  In this work, our objective is to find out how topological and algebraic
properties of unrooted Gaussian tree models determine their security
robustness, which is measured by our proposed max-min information (MaMI)
metric. Such metric quantifies the amount of common randomness extractable
through public discussion between two legitimate nodes under an eavesdropper
attack. We show some general topological properties that the desired max-min
solutions shall satisfy. Under such properties, we develop conditions under
which comparable trees are put together to form partially ordered sets
(posets). Each poset contains the most favorable structure as the poset leader,
and the least favorable structure. Then, we compute the Tutte-like polynomial
for each tree in a poset in order to assign a polynomial to any tree in a
poset. Moreover, we propose a novel method, based on restricted integer
partitions, to effectively enumerate all poset leaders. The results not only
help us understand the security strength of different Gaussian trees, which is
critical when we evaluate the information leakage issues for various jointly
Gaussian distributed measurements in networks, but also provide us both an
algebraic and a topological perspective in grasping some fundamental properties
of such models.
