  For a specific class of sparse Gaussian graphical models, we provide a
closed-form solution for the determinant of the covariance matrix. In our
framework, the graphical interaction model (i.e., the covariance selection
model) is equal to replacement product of $\mathcal{K}_{n}$ and
$\mathcal{K}_{n-1}$, where $\mathcal{K}_n$ is the complete graph with $n$
vertices. Our analysis is based on taking the Fourier transform of the local
factors of the model, which can be viewed as an application of the Normal
Factor Graph Duality Theorem and holographic algorithms. The closed-form
expression is obtained by applying the Matrix Determinant Lemma on the
transformed graphical model. In this context, we will also define a notion of
equivalence between two Gaussian graphical models.
