Abstract
We continue the discussion of the fermion models on graphs that started in the first paper
of the series. Here we introduce a graphical gauge model (GGM) and show that: (a) it can
be stated as an average/sum of a determinant defined on the graph over a 
(binary) gauge field; (b) it is equivalent to the monomer–dimer (MD) model on the graph;
(c) the partition function of the model allows an explicit expression in terms of a series over
disjoint directed cycles, where each term is a product of local contributions along the cycle
and the determinant of a matrix defined on the remainder of the graph (excluding the
cycle). We also establish a relation between the MD model on the graph and the
determinant series, discussed in the first paper—however, considered using simple
non-belief propagation choice of the gauge. We conclude with a discussion of possible
analytic and algorithmic consequences of these results, as well as related questions and
challenges.