In this thesis, we leverage finite graph covers to analyze the SPA and the Bethe partition function for both S-FGs and DE-FGs. There are two main contributions in this thesis. The first main contribution concerns a special class of S-FGs where the partition function of each S-FG equals the permanent of a nonnegative square matrix. The Bethe partition function for such an S-FG is called the Bethe permanent. A combinatorial characterization of the Bethe permanent is given by the degree-$M$ Bethe permanent, which is defined based on the degree-$M$ graph covers of the underlying S-FG. In this thesis, we prove a degree-$M$-Bethe-permanent-based lower bound on the permanent of a non-negative square matrix, resolving a conjecture proposed by Vontobel in [IEEE Trans. Inf. Theory, Mar. 2013]. We also prove a degree-$M$-Bethe-permanent-based upper bound on the permanent of a non-negative matrix. In the limit $M \to \infty$, these lower and upper bounds yield known Bethe-permanent-based lower and upper bounds on the permanent of a non-negative square matrix. The second main contribution is giving a combinatorial characterization of the Bethe partition function for DE-FGs in terms of finite graph covers. In general, approximating the partition function of a DE-FG is more challenging than for an S-FG because the partition function of the DE-FG is a sum of complex values and not just a sum of non-negative real values. Moreover, one cannot apply the method of types for proving the combinatorial characterization as in the case of S-FGs. We overcome this challenge by applying a suitable loop-calculus transform (LCT) for both S-FGs and DE-FGs. Currently, we provide a combinatorial characterization of the Bethe partition function in terms of finite graph covers for a class of DE-FGs satisfying an (easily checkable) condition.

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