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

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generation of Farey-Type Graphs

**Definition**

**1.**

- $F(0)$has two vertices and an edge.
- For$t\ge 1$,$F(t)$is obtained from$F(t-1)$by adding a new vertex to every edge which is introduced at step$t-1$in$F(t-1)$, then to linking the new vertex to the two vertices of that edge.

**Remark**

**1.**

**Definition**

**2.**

- For$t=0$,$GF(0,k)$is composed of three initial vertices which are linked to each other.
- For$t\ge 1$,$GF(t,k)$is constructed from$GF(t-1,k)$by adding$k$new vertices to every edge introduced at step$t-1$, then linking the$k$new vertices to the two end vertices.

**Remark**

**2.**

**Definition**

**3.**

- For$t=0$, $EF(0,k)$ holds three vertices that are linked to each other.
- For$t\ge 1$,$EF(t,k)$is constructed from$EF(t-1,k)$by adding$k$new vertices to every edge linked at step$t-1$and three initial edges added at$t=0$, then linking the$k$new vertices to the two end vertices (see Figure 4).

**Remark**

**3.**

## 3. Labeling and Routing of $F(t)$

**Definition**

**4.**

- Label two initial vertices as$0.1$and$0.0$when t = 0.
- When$t\ge 1$, the new vertices added at step t are marked with labels from$t.1$to$t{.2}^{t-1}$in a clockwise direction.

**Property**

**1.**

**Property**

**2.**

**Property**

**3.**

**Property**

**4.**

**Property**

**5.**

- 1.
- Given a pair of vertices labeled with${t}_{i}.k$and${t}_{j}.l$.
- 2.
- Determine whether the two vertices are neighbors or not.

**Property**

**6.**

**Proof.**

## 4. Labeling of $GF(t,k)$ and $EF(t,k)$

**Definition**

**5.**

- The three initial vertices are labeled with 0, 1 and 2.
- At any step$t\ge 1$, 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$a\in \{0,1,2\}$,$b=\{1,2,\dots ,t\}$,$c\in \{1,2,\dots ,{2}^{b-1}\}$and$d\in \{1,2,\dots ,{k}^{b}\}$.

**Definition**

**6.**

- Label three initial vertices as 0, 1 and 2.
- At step$t\ge 1$, a vertex is tagged with$a.b.c.d$, in which$a\in \{0,1,2\}$,$b=\{1,2,\dots ,t\}$,$c\in \{1,2,\dots ,$${2}^{b-1}\}$and$d\in \{1,2,\dots ,(t-b+1)\times {k}^{b}\}$.

**Remark**

**4.**

**Property**

**7.**

**Proof.**

**Remark**

**5.**

**Property**

**8.**

**Property**

**9.**

**Proof.**

**Property**

**10.**

**Proof.**

## 5. Routing of $GF(t,k)$ and $EF(t,k)$

**Property**

**11.**

**Proof.**

**Property**

**12.**

**Proof.**

**Property**

**13.**

**Proof.**

**Property**

**14.**

**Remark**

**6.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Comellas, F.; Ozon, J.; Peters, J.G. Deterministic small-world communication networks. Inf. Process. Lett.
**2000**, 76, 83–90. [Google Scholar] [CrossRef] - Barabási, A.L.; Ravasz, E.; Vicsek, T. Deterministic scale-free networks. Phys. A Stat. Mech. Appl.
**2001**, 299, 559–564. [Google Scholar] [CrossRef] [Green Version] - Perc, M. The Matthew effect in empirical data. J. R. Soc. Interface
**2014**, 11, 20140378. [Google Scholar] [CrossRef] [PubMed] - Jalili, M.; Perc, M. Information cascades in complex networks. J. Complex Netw.
**2017**, 5, 665–693. [Google Scholar] [CrossRef] - Zhang, Z.; Gao, S.; Chen, L.; Zhou, S.; Zhang, H.; Guan, J. Mapping Koch curves into scale-free small-world networks. J. Phys. A Math. Theor.
**2010**, 43, 395101. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z.; Rong, L.; Zhou, S. Evolving Apollonian networks with small-world scale-free topologies. Phys. Rev. E
**2006**, 74, 046105. [Google Scholar] [CrossRef] [PubMed] - Zhou, T.; Wang, B.H.; Hui, P.M.; Chan, K.P. Topological properties of integer networks. Phys. A Stat. Mech. Appl.
**2006**, 367, 613–618. [Google Scholar] [CrossRef] [Green Version] - Perc, M.; Jordan, J.J.; Rand, D.G.; Wang, Z.; Boccaletti, S.; Szolnoki, A. Statistical physics of human cooperation. Phys. Rep.
**2017**, 687, 1–51. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.; Bauch, C.T.; Bhattacharyya, S.; d’Onofrio, A.; Manfredi, P.; Perc, M.; Perra, N.; Salathe, M.; Zhao, D. Statistical physics of vaccination. Phys. Rep.
**2016**, 664, 1–113. [Google Scholar] [CrossRef] [Green Version] - D’Orsogna, M.R.; Perc, M. Statistical physics of crime: A review. Phys. Life Rev.
**2015**, 12, 1–21. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Helbing, D.; Brockmann, D.; Chadefaux, T.; Donnay, K.; Blanke, U.; Woolley-Meza, O.; Moussaid, M.; Johansson, A.; Krause, J.; Schutte, S.; et al. Saving human lives: What complexity science and information systems can contribute. J. Stat. Phys.
**2015**, 158, 735–781. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z.; Comellas, F. Farey graphs as models for complex networks. Theor. Comput. Sci.
**2011**, 412, 865–875. [Google Scholar] [CrossRef] - Zhang, Z.; Wu, B.; Lin, Y. Counting spanning trees in a small-world Farey graph. Phys. A Stat. Mech. Appl.
**2012**, 391, 3342–3349. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z.; Rong, L.; Guo, C. A deterministic small-world network created by edge iterations. Phys. A Stat. Mech. Appl.
**2006**, 363, 567–572. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z.Z.; Rong, L.L.; Comellas, F. Evolving small-world networks with geographical attachment preference. J. Phys. A Math. Gen.
**2006**, 39, 3253. [Google Scholar] [CrossRef] - Peng, A.; Zhang, L. Deterministic multidimensional growth model for small-world networks. arXiv, 2011; arXiv:1108.5450. [Google Scholar]
- Zhang, Z.; Rong, L.; Zhou, S. A general geometric growth model for pseudofractal scale-free web. Phys. A Stat. Mech. Appl.
**2007**, 377, 329–339. [Google Scholar] [CrossRef] [Green Version] - Havlin, S.; ben-Avraham, D. Fractal and transfractal recursive scale-free nets. New J. Phys.
**2007**, 9, 175. [Google Scholar] - Xiao, Y.; Zhao, H. Counting the number of spanning trees of generalization Farey graph. In Proceedings of the 2013 Ninth International Conference on Natural Computation (ICNC), Shenyang, China, 23–25 July 2013; pp. 1778–1782. [Google Scholar]
- Andrade, J.S., Jr.; Herrmann, H.J.; Andrade, R.F.; Da Silva, L.R. Apollonian networks: Simultaneously scale-free, small world, Euclidean, space filling, and with matching graphs. Phys. Rev. Lett.
**2005**, 94, 018702. [Google Scholar] [CrossRef] [PubMed] - Auto, D.M.; Moreira, A.A.; Herrmann, H.J.; Andrade, J.S., Jr. Finite-size effects for percolation on Apollonian networks. Phys. Rev. E
**2008**, 78, 066112. [Google Scholar] [CrossRef] [PubMed] - Almeida, G.M.; Souza, A.M. Quantum transport with coupled cavities on an Apollonian network. Phys. Rev. A
**2013**, 87, 033804. [Google Scholar] [CrossRef] - Wong, W.K.; Guo, Z.X.; Leung, S.Y.S. Partially connected feedforward neural networks on Apollonian networks. Phys. A Stat. Mech. Appl.
**2010**, 389, 5298–5307. [Google Scholar] [CrossRef] - Mendes, G.A.; Da Silva, L.R.; Herrmann, H.J. Traffic gridlock on complex networks. Phys. A Stat. Mech. Appl.
**2012**, 391, 362–370. [Google Scholar] [CrossRef] - De Oliveira, I.N.; de Moura, F.A.B.F.; Lyra, M.L.; Andrade, J.S., Jr.; Albuquerque, E.L. Bose-Einstein condensation in the Apollonian complex network. Phys. Rev. E
**2012**, 81, 030104. [Google Scholar] [CrossRef] [PubMed] - De Oliveira, I.N.; De Moura, F.A.B.F.; Lyra, M.L.; Andrade, J.S., Jr.; Albuquerque, E.L. Free-electron gas in the Apollonian network: Multifractal energy spectrum and its thermodynamic fingerprints. Phys. Rev. E
**2009**, 79, 016104. [Google Scholar] [CrossRef] [PubMed] - Knuth, D.E. A generalization of Dijkstra’s algorithm. Inf. Process. Lett.
**1977**, 6, 1–5. [Google Scholar] [CrossRef] - Yen, J.Y. An algorithm for finding shortest routes from all source nodes to a given destination in general networks. Q. Appl. Math.
**1970**, 27, 526. [Google Scholar] [CrossRef] - Lerner, J.; Wagner, D.; Zweig, K. Engineering route planning algorithms. In Algorithmics of Large and Complex Networks; Springer: Berlin/Heidelberg, Germany, 2009; pp. 117–139. ISBN 978-3-642-02093-3. [Google Scholar]
- Zwick, U. All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM
**2002**, 49, 289–317. [Google Scholar] [CrossRef] [Green Version] - Chan, T.M. More algorithms for all-pairs shortest paths in weighted graphs. SIAM J. Comput.
**2010**, 39, 2075–2089. [Google Scholar] [CrossRef] - Yen, J.Y. Finding the k shortest loopless paths in a network. Manag. Sci.
**1971**, 17, 712–716. [Google Scholar] [CrossRef] - Bern, M.W.; Graham, R.L. The shortest-network problem. Sci. Am.
**1989**, 260, 84–89. [Google Scholar] [CrossRef] - Zwick, U. Exact and approximate distances in graphs—A survey. In Algorithms—ESA 2001; Springer: Berlin/Heidelberg, Germany, 2001; pp. 33–48. [Google Scholar]
- Jiang, W.; Zhai, Y.; Martin, P.; Zhao, Z. Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks. Sci. Rep.
**2018**, 8, 12194. [Google Scholar] [CrossRef] [PubMed] - Comellas, F.; Miralles, A. Vertex labeling and routing in self-similar outerplanar unclustered graphs modeling complex networks. J. Phys. A Math. Theor.
**2009**, 42, 425001. [Google Scholar] [CrossRef] [Green Version] - Comellas, F.; Miralles, A. Label-based routing for a family of scale-free, modular, planar and unclustered graphs. J. Phys. A Math. Theor.
**2011**, 44, 205102. [Google Scholar] [CrossRef] - Comellas, F.; Fertin, G.; Raspaud, A. Vertex Labeling and Routing in Recursive Clique-Trees, a New Family of Small-World Scale-Free Graphs. In Proceedings of the SIROCCO 2003, Umeå, Sweden, 18–20 June 2003; pp. 73–87. [Google Scholar]
- Zhai, Y.; Wang, Y. Label-based routing for a family of small-world Farey graphs. Sci. Rep.
**2016**, 6, 25621. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ball, F.; Geyer-Schulz, A. How symmetric are real-world graphs? A large-scale study. Symmetry
**2018**, 10, 29. [Google Scholar] [CrossRef] - Parada, R.; Cárdenes-Tacoronte, D.; Monzo, C.; Melià-Seguí, J. Internet of THings Area Coverage Analyzer (ITHACA) for complex topographical scenarios. Symmetry
**2017**, 9, 237. [Google Scholar] [CrossRef] - Garrido, A. Symmetry in complex networks. Symmetry
**2011**, 3, 1–15. [Google Scholar] [CrossRef] - Garlaschelli, D.; Ruzzenenti, F.; Basosi, R. Complex networks and symmetry I: A review. Symmetry
**2010**, 2, 1683–1709. [Google Scholar] [CrossRef] - Zhang, Y.; Zhang, Z.; Zhou, S.; Guan, J. Deterministic weighted scale-free small-world networks. Phys. A Stat. Mech. Appl.
**2010**, 389, 3316–3324. [Google Scholar] [CrossRef] [Green Version] - Sun, W.; Wu, Y.; Chen, G.; Wang, Q. Deterministically delayed pseudofractal networks. J. Stat. Mech. Theory Exp.
**2011**, 10, P10032. [Google Scholar] [CrossRef]

**Figure 3.**The generalization of Farey graphs. (

**a**) $GF(t,k)$ at steps t = 0, 1 and 2 when $k=1$; (

**b**) $GF(t,k)$ at steps t = 0, 1 and 2 when $k=2$.

**Figure 4.**The extended Farey graphs. (

**a**) $EF(t,k)$ at steps t = 0, 1 and 2 when $k=1$; (

**b**) $EF(t,k)$ at steps t = 0, 1 and 2 when $k=2$.

**Figure 6.**The labeling of $GF(t,k)$ and $EF(t,k)$ at step t = 2 for $k=2$. (

**a**) $GF(2,2)$; (

**b**) $EF(2,2)$.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jiang, W.; Zhai, Y.; Zhuang, Z.; Martin, P.; Zhao, Z.; Liu, J.-B.
Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs. *Symmetry* **2018**, *10*, 407.
https://doi.org/10.3390/sym10090407

**AMA Style**

Jiang W, Zhai Y, Zhuang Z, Martin P, Zhao Z, Liu J-B.
Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs. *Symmetry*. 2018; 10(9):407.
https://doi.org/10.3390/sym10090407

**Chicago/Turabian Style**

Jiang, Wenchao, Yinhu Zhai, Zhigang Zhuang, Paul Martin, Zhiming Zhao, and Jia-Bao Liu.
2018. "Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs" *Symmetry* 10, no. 9: 407.
https://doi.org/10.3390/sym10090407