  In analogy with the Manin-Mumford conjecture for algebraic curves, one may
ask how a metric graph under the Abel-Jacobi embedding intersects torsion
points of its Jacobian. We show that the number of torsion points is finite for
metric graphs of genus $g\geq 2$ which are biconnected and have edge lengths
which are "sufficiently irrational" in a precise sense. Under these
assumptions, the number of torsion points is bounded by $3g-3$. Next we study
bounds on the number of torsion points in the image of higher-degree
Abel-Jacobi embeddings, which send $d$-tuples of points to the Jacobian. This
motivates the definition of the "independent girth" of a graph, a number which
is a sharp upper bound for $d$ such that the higher-degree Manin-Mumford
property holds.
