  Metro networks serve as good examples of traffic systems for understanding
the relations between geometric structures and transport properties.We study
and compare 28 world major metro networks in terms of the Wasserstein distance,
the key metric for optimal transport, and measures geometry related, e.g.
fractal dimension, graph energy and graph spectral distance. The finding of
power-law relationships between rescaled graph energy and fractal dimension for
both unweighted and weighted metro networks indicates the energy costs per unit
area are lower for higher dimensioned metros. In L space, the mean Wasserstein
distance between any pair of connected stations is proportional to the fractal
dimension, which is in the vicinity of our theoretical calculations treated on
special regular tree graphs. This finding reveals the geometry of metro
networks and tree graphs are in close proximity to one another. In P space, the
mean Wasserstein distance between any pair of stations relates closely to the
average number of transfers. By ranking several key quantities transport
concerned, we obtain several ranking lists in which New York metro and Berlin
metro consistently top the first two spots.
