Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs

The generalization of Farey graphs and extended Farey graphs are all originated from Farey graph and scale-free and small-world simultaneously. We propose a labeling of the vertices for it that allows determining all the shortest paths routing between any two vertices only based on their labels. The maximum number of shortest paths between any two vertices is huge as the product of two Fibonacci numbers, however, the label-based routing algorithm runs in linear time O(n). The existence of an efficient routing protocol for Farey-type models should help the understanding of several physical dynamic processes on it.