# Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs
## 1 Introduction  
In comparison with empirical and random graphs, deterministic models have unique advantages in improving our comprehension in complex networks. For instance, we can obtain the solutions for a deterministic model by rigorous derivation, and the computation is only a small quantity of calculations. Deterministic models can be created by simple recursive operation [1; 2; 3; 4], construction of fractal processes [5], plane filling techniques [6], the relationships between natural numbers [7], or perspectives and methods from classical research of physical processes [8; 9; 10; 11]. Recently, [PERSON] et al. introduced Farey graphs (FG) on the basis of classical Farey sequences. Farey graphs are simultaneously uniquely Hamiltonian, minimally three-colorable, maximally outer-planar, and perfect [12; 13]. FG can be used as modules of multiple networks. The networks created by edge iterations [14], or evolving graphs with geographical attachment preference [15], coincide with the combination of three FG, while the merger of six FG generates graphs with multidimensional growth [16]. Moreover, two new kinds of Farey-type graphs, the generalization of Farey graphs (GFG), and the extended Farey graphs (EFG) are deduced by generalizing the construction mechanism of FG, and they all are scale-free and small-world [17; 18; 19]. Deterministic graphs also provide a new perspective and methodology for classical research of physical processes. For example, some important dynamical processes on the basis of Apollonian models [20], a kind of deterministic matching graph, space-filling, Euclidean, small-world, and scale-free,has been researched extensively. Accurate analytical solutions are derived, including percolation [21], electrical conduction [20], Ising models [20], quantum transport [22], partially connected feedforward neural networks [23], traffic gridlock [24], Bose-Einstein condensation [25], free-electron gas [26], and so on. However, the study of physical processes using FG is still lacking, as the recursive relationships in FG are more complex than for Apollonian networks. The label-based routing protocol for all shortest paths may also bridge Farey-type graphs with the fields below. The all-pairs shortest paths problem is frequently studied in textbooks and it has remained open to this day, the classical algorithms such as the A*, Bellman-Ford, Floyd-Warshall and Dijkstra algorithms all run in subcubic time \(O(\left|V\right|^{3-\delta})\), where \(\delta>0\)[27; 28; 29; 30; 31]. The \(K\) shortest path routing (KSPR) algorithm is an extension of the shortest path routing algorithm, in which \(K\) is the number of shortest paths to find [32]. KSPR finds \(K\) paths in order of increasing cost, including the shortest path. KSPR in Farey-type graphs will partly reduce to finding out all the shortest paths. Similar to the minimum spanning problem, the graph Steiner tree problem (GSTR) interconnects a set \(V\) of vertices by using a network of shortest length, and the length is the sum of the lengths of all edges [33]. Most versions of GSTR are NP (non-deterministic polynomial) complete. Moreover, the bottleneck of many network analysis algorithms is the exorbitant computational complexity of calculating the shortest paths, so that scientists are forced to use approximation algorithms [34]. Graph labeling is one active subject in graph theory. Graph labeling has a wide range of applications in many fields, such as communication design, circuit design, crystallography, and coding theory [35]. Finding the shortest paths in graphs is an open problem, which has been well studied. [PERSON] and [PERSON] pioneered the relationship between the shortest paths and vertex labels [35; 36; 37; 38; 39]. They can determine the shortest path just by applying simple rules and a few computations based only on their labels. We propose an algorithm for labeling nodes--all of the shortest paths between any pair of vertices can be efficiently derived only on the basis of their labels.