  This is a sequel to [arXiv:1708.09092v2]. For an oriented trivalent graph $G$
without source or sink embedded in $S^3$, we prove that the $\mathfrak{gl}(1|
1)$-Alexander polynomial $\underline{\Delta}(G, c)$ defined by Viro satisfies a
series of relations, which we call MOY-type relations in [arXiv:1708.09092v2].
As a corollary we show that the Alexander polynomial $\Delta_{(G, c)}(t)$
studied in [arXiv:1708.09092v2] coincides with $\underline{\Delta}(G, c)$ for a
positive coloring $c$ of $G$, where $\Delta_{(G, c)}(t)$ is constructed from
certain regular covering space of the complement of $G$ in $S^3$ and it is the
Euler characteristic of the Heegaard Floer homology of $G$ that we studied
before. When $G$ is a plane graph, we provide a topological interpretation to
the vertex state sum of $\underline{\Delta}(G, c)$ by considering a special
Heegaard diagram of $G$ and the Fox calculus on the Heegaard surface.
