First, let me explain, what I understand by a "Cantor Graph":
it is an infinite, directed graph with self loops and countably many vertices labelled with the natural numbers; every ordered pair of vertices is connected by an arc, whose weight equals the quotient of the labels of its tail and head vertex (the graph is inspired by Cantor's famous proof of the countability of the rationals).
Now, my questions are, given two natural numbers, $m$ and $n$,
1.) how long is the shortest path from $m$ to $n$? and,
2.) how many arcs are on the shortest path from $m$ to $n$?
3.) can the above questions be answered without partially constructing the "Cantor Graph"?
Going, for example, directly from $10$ to $2$ would result in the length of the arc from $10$ to $2$, whose length is 5; going via $5$ would result in a path length of 10/5 + 5/2 = 9/2 and is thus shorter than going via the direct connection.