  In this paper, we study the Hankel edge ideals of graphs. We determine the
minimal prime ideals of the Hankel edge ideal of labeled Hamiltonian and
semi-Hamiltonian graphs, and we investigate radicality, being a complete
intersection, almost complete intersection and set theoretic complete
intersection for such graphs. We also consider the Hankel edge ideal of trees
with a natural labeling, called rooted labeling. We characterize such trees
whose Hankel edge ideal is a complete intersection, and moreover, we determine
those whose initial ideal with respect to the reverse lexicographic order
satisfies this property.
