  Metric graphs are meaningful objects for modeling complex structures that
arise in many real-world applications, such as road networks, river systems,
earthquake faults, blood vessels, and filamentary structures in galaxies. To
study metric graphs in the context of comparison, we are interested in
determining the relative discriminative capabilities of two topology-based
distances between a pair of arbitrary finite metric graphs: the persistence
distortion distance and the intrinsic Cech distance. We explicitly show how to
compute the intrinsic Cech distance between two metric graphs based solely on
knowledge of the shortest systems of loops for the graphs. Our main theorem
establishes an inequality between the intrinsic Cech and persistence distortion
distances in the case when one of the graphs is a bouquet graph and the other
is arbitrary. The relationship also holds when both graphs are constructed via
wedge sums of cycles and edges.
