  In our previous paper we associated to each non-constant elliptic function
$f$ on a torus $T$ a dynamical system, the elliptic Newton flow corresponding
to $f$. We characterized the functions for which these flows are structurally
stable and showed a genericity result. In the present paper we focus on the
classification and representation of these structurally stable flows. The phase
portrait of a structurally stable elliptic Newton flow generates a connected,
cellularly embedded, graph $\mathcal{G}(f)$ on a torus $T$ with $r$ vertices,
2$r$ edges and $r$ faces that fulfil certain combinatorial properties ( Euler,
Hall) on some of its subgraphs. The graph $\mathcal{G}(f)$ determines the
conjugacy class of the flow. [classification] A connected, cellularly embedded
toroidal graph $\mathcal{G}$ with the above Euler and Hall properties, is
called a Newton graph. Any Newton graph $\mathcal{G}$ can be realized as the
graph $\mathcal{G}(f)$ of the structurally stable Newton flow for some function
$f$. This leads to: up till conjugacy between flows and (topological)
equivalency between graphs, there is a one to one correspondence between the
structurally stable Newton flows and Newton graphs, both with respect to the
same order $r$ of the underlying functions $f$.[representation] Finally, we
clarify the analogy between rational and elliptic Newton flows, and show that
the detection of elliptic Newton flows is possible in polynomial time.
