# Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs
## 4 Labeling of \(GF(t,k)\) and \(EF(t,k)\)  
**Definition 5**. : _(The labeling of GFG) The labeling of any vertex in \(GF(t,k)\) adheres to the following rules:_  
* _The three initial vertices are labeled with 0, 1 and 2._
* _At any step_ \(t\geq 1\)_, a vertex in_ \(GF(t,k)\) _is marked with a.b.c.d according to the group ( a ), the subgroup ( b.c ) and the precise positions (d) from down to top in the same subgroup, in which_ \(a\in\{0,1,2\}\)_,_ \(b=\{1,2,\ldots,t\}\)_,_ \(c\in\left\{1,2,\ldots,2^{b-1}\right\}\) _and_ \(d\in\left\{1,2,\ldots,k^{b}\right\}\)_._  
**Definition 6**. : _(The labeling of EFG) Vertices in \(EF(t,k)\) are labeled as follows:_  
* _Label three initial vertices as 0, 1 and 2._
* _At step_ \(t\geq 1\)_, a vertex is tagged with a.b.c.d, in which_ \(a\in\{0,1,2\}\)_,_ \(b=\{1,2,\ldots,t\}\)_,_ \(c\in\{1,2,\ldots,\)__\(2^{b-1}\}\) _and_ \(d\in\left\{1,2,\ldots,(t-b+1)\times k^{b}\right\}\)_.__Labeling of_ \(GF(2,2)\) _and_ \(EF(2,2)\) _are illustrated in Figure_ 6_._  
Figure 6: The labeling of \(GF(t,k)\) and \(EF(t,k)\) at step \(t=2\) for \(k=2\). (**a**) \(GF(2,2)\); (**b**) \(EF(2,2)\). **Remark 4**. : _A GFG has more petals than an EFG, such that \(d_{\max}=(t-b+1)\times k^{b}\) in EFG, while \(d_{\max}=k^{b}\) in GFG._  
When \(t\geq 2\), any new vertex, added to GFG/EFG at step \(t_{i}\), links to two vertices: a mother and a father. The vertices added to graphs at the same time are siblings if they have the same parents. Supposing that two arbitrary vertices in GFG/EFG are labeled with \(a_{1}.b_{1}.c_{1}.d_{1}\) and \(a_{2}.b_{2}.c_{2}.d_{2}\), in which \(b_{1}\geq b_{2}\), then, we give several properties satisfying GFG and EFG at once. **Property 7**. : _(The family of a.b.c.d)_  
_When \(b\geq 1\), a.b.c.d belongs to a set of siblings \(\{a.b.c.(k\Big{\lfloor}\frac{d}{k}\Big{\rfloor}+1),a.b.c. (k\Big{\lfloor}\frac{ d}{k}\Big{\rfloor}+2),\ldots,a.b.c. (k\Big{\lfloor}\frac{d}{k}\Big{\rfloor}+k)\}\)._  
_When \(b\geq 2\), a.b.c.d and its parents define a triangle, the mother is \(a. (b-1).\big{\lfloor}\frac{c}{k}\big{\rfloor}\), the father is \(a. (b-l).\Big{\lfloor}\frac{c-rem(c,2)}{2^{l}}\Big{\rfloor}.\Big{\lfloor}\frac {d}{k}\Big{\rfloor}\)._  
_If \(b\geq b_{0}\), the \((b-b_{0})\)th generation of maternal ancestor of \(a.b.c.d\) is \(a.b_{0}.\Big{\lfloor}\frac{c}{2^{b-b_{0}}}\Big{\rfloor}.\Big{\lfloor}\frac{d} {k^{b-b_{0}}}\Big{\rfloor}\)._  
**Proof. ** The above results are self-evident, with the exception of the formulation of the father's label, which we shall now prove. If \(b\geq 2\), let \(l\) denotes the difference \(b-b_{1}\), thus, \(l\in\{2,\ldots,b-2\}\). When \(c=1\) and with any \(b\), the fathers are all the initial vertexes. When \(c\) is odd but excluding one, \(l\) is one plus the number of the continuous zeros from right to left in the binary representation of \(c-1\), such that the father is labeled as \(a. (b-l).\Big{\lfloor}\frac{c-1}{2^{l}}\Big{\rfloor}.\Big{\lfloor}\frac{d}{k^{ l}}\Big{\rfloor}\). When \(c\) is even, the time difference \(l\) is one plus the number of the continuous zeros from right to left of the binary representation of \(c\), such that the father's label is \(a. (b-l).\Big{\lfloor}\frac{c}{2^{l}}\Big{\rfloor}.\Big{\lfloor}\frac{d}{k^{ l}}\Big{\rfloor}\). In summary, the father of \(a.b.c.d\) is \(a. (b-l).\Big{\lfloor}\frac{c-rem(c,2)}{2^{l}}\Big{\rfloor}.\Big{\lfloor}\frac {d}{k^{l}}\Big{\rfloor}\) when \(b\geq 2\). \(\Box\)  
**Remark 5**. : _The vertex \(a.1.1.d\) has two mothers \(\{0,1,2\}/\{a\}\) and no father. The three initial vertices have no parents or siblings. Therefore, the neighbors of \(a.b.c.d\) are derived from Property 8._  
**Property 8**. : _(The neighbors of a.b.c.d)_  
_When \(b\geq 2\), the set of neighbors of \(a.b.c.d\) is \(\Big{\{}a. (b-l).\Big{\lfloor}\frac{c-rem(c,2)}{2^{l}}\Big{\rfloor}.\Big{ \lfloor}\frac{d}{k^{l}}\Big{\rfloor},a. (b-1).\big{\lfloor}\frac{c}{k}\big{\rfloor}.\Big{\lfloor}\frac{d}{k}\Big{\rfloor}\), \(a. (b+x).2^{x-1}(2c-1).d\), \(a.(b+x).