The edge-Wiener index is an important topological index in Chemical Graph Theory, defined as the sum of distances among all pairs of edges. Fractal structures have received much attention from scientists because of their philosophical and aesthetic significance, and chemists have even attempted to synthesize various types of molecular fractal structures. The level-3 Sierpinski triangle is constructed similarly to the Sierpinski triangle and its skeleton networks have self-similarity. In this paper, by using the method of finite pattern, we obtain the edge-Wiener index of skeleton networks according to level-3 Sierpinski triangle. This provides insights for a better understanding of molecular fractal structures.

Keywords:

References

1. D. J. Watts and S. H. Strogatz , Collective dynamics of ‘small-world’ networks, Nature 393 (1998) 440–442. Crossref, Web of Science, Google Scholar
2. A. L. Barabási and R. Albert , Emergence of scaling in random networks, Science 286 (1999) 509–512. Crossref, Web of Science, Google Scholar
3. C. Song, S. Havlin and H. A. Makse , Self-similarity of complex networks, Nature 433(7024) (2005) 392–395. Crossref, Web of Science, Google Scholar
4. C. Song, S. Havlin and H. A. Makse , Origins of fractality in the growth of complex networks, Nat. Phys. 2(4) (2006) 275–281. Crossref, Web of Science, Google Scholar
5. C. Song, L. K. Gallos, S. Havlin and H. A. Makse , How to calculate the fractal dimension of a complex network: The box covering algorithm, J. Stat. Mech. Theory Exp. 2007(03) (2007) P03006. Crossref, Web of Science, Google Scholar
6. P. Xie, Y. Lin and Z. Zhang , Spectrum of walk matrix for Koch network and its application, J. Chem. Phys. 142(22) (2015) 224106. Crossref, Web of Science, Google Scholar
7. Z. Zhang, S. Zhou, Z. Su, T. Zou and J. Guan , Random Sierpinski network with scale-free small-world and modular structure, Eur. Phys. J. B 65 (2008) 141–147. Crossref, Web of Science, Google Scholar
8. M. Dai, Y. Zong, J. He et al., The trapping problem of the weighted scale-free treelike networks for two kinds of biased walks, Chaos 28(11) (2018) 113115. Crossref, Web of Science, Google Scholar
9. M. Dai, J. He, Y. Zong et al., Coherence analysis of a class of weighted networks, Chaos 28 (2018) 043110. Crossref, Web of Science, Google Scholar
10. C. Zeng, Y. Xue and M. Zhou , Fractal networks on Sierpinski-type polygon, Fractals 28(05) (2020) 2050087. Link, Web of Science, Google Scholar
11. Y. Huang, H. Zhang, C. Zeng et al., Scale-free and small-world properties of a multiple-hub network with fractal structure, Physica A 558 (2020) 125001. Crossref, Web of Science, Google Scholar
12. J. Shang, Y. Wang, M. Chen et al., Assembling molecular Sierpinski triangle fractals, Nat. Chem. 7(5) (2015) 389–393. Crossref, Web of Science, Google Scholar
13. Y. Mo, T. Chen, J. Dai et al., On-surface synthesis of highly ordered covalent Sierpinski triangle fractals, J. Am. Chem. Soc. 141(29) (2019) 11378–11382. Crossref, Web of Science, Google Scholar
14. J. Dai, X. Zhao, Z. Peng et al., Assembling surface molecular Sierpinski triangle fractals via K $^{+}$ -invoked electrostatic interaction, J. Am. Chem. Soc. 145(25) (2023) 13531–13536. Crossref, Web of Science, Google Scholar
15. C. Li, R. Li, Z. Xu et al., Packing biomolecules into Sierpinski triangles with global organizational chirality, J. Am. Chem. Soc. 143(36) (2021) 14417–14421. Crossref, Web of Science, Google Scholar
16. D. Bonchev and D. H. Rouvray (eds.), Chemical Graph Theory: Introduction and Fundamentals (Taylor & Francis, 1991). Google Scholar
17. H. Wiener , Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69(1) (1947) 17–20. Crossref, Web of Science, Google Scholar
18. I. Gutman and N. Trinajstić , Graph theory and molecular orbitals. Total $π$ -electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17(4) (1972) 535–538. Crossref, Web of Science, Google Scholar
19. M. Randić , Characterization of molecular branching, J. Am. Chem. Soc. 97(23) (1975) 6609–6615. Crossref, Web of Science, Google Scholar
20. H. Mujahed and R. Basbous , Java-based program for computing the Wiener index of a body-centered cubic graph, Palestinian J. Technol. Appl. Sci. 1(1) (2018) 50–57. Google Scholar
21. A. Mahboob and M. Rasheed , Hosaya polynomial and Weiner index of Abid–Waheed graph (AW) $_{p}^{6}$ , Appl. Comput. Math. 10(3) (2022) 89–94. Google Scholar
22. A. Iranmanesh, I. Gutman, O. Khormali and A. Mahmiani , The edge versions of Wiener index, MATCH Commun. Math. Comput. Chem. 61 (2009) 663–672. Web of Science, Google Scholar
23. P. Dankelmann, I. Gutman, S. Mukwembi and H. Swart , The edge-Wiener index of a graph, Discrete Math. 309(10) (2009) 3452–3457. Crossref, Web of Science, Google Scholar
24. K. Cui and L. Xi , The hyper-Wiener indices of Vicsek networks, Fractals 30(9) (2022) 2250181. Link, Web of Science, Google Scholar
25. Y. Lu, J. Xu and L. Xi , Fractal version of hyper-Wiener index, Chaos, Solitons Fractals 166 (2023) 112973. Crossref, Web of Science, Google Scholar
26. J. Gu, J. Fan, Q. Ye and L. Xi , Mean geodesic distance of the level- $n$ Sierpinski gasket, J. Math. Anal. Appl. 508(1) (2022) 125853. Crossref, Web of Science, Google Scholar
27. J. Hutchinson , Fractals and self similarity, Indiana Univ. Math. J. 30(5) (1981) 713–747. Crossref, Web of Science, Google Scholar
28. S. Wang, Z. Yu and L. Xi , Average geodesic distance of Sierpinski gasket and Sierpinski networks, Fractals 25(5) (2017) 1750044. Link, Web of Science, Google Scholar
29. B. Zhao, J. Fan and L. Xi , Average distances of Lindström snowflake networks, Fractals 29(3) (2021) 2150067. Link, Web of Science, Google Scholar
30. Y. Li, J. Fan and L. Xi , Average geodesic distance on stretched Sierpinski gasket, Chaos Solitons Fractals 150 (2021) 111120. Crossref, Web of Science, Google Scholar
31. J. Yang, S. Wang, L. Xi and Y. Ye , Average geodesic distance of skeleton networks of Sierpinski tetrahedron, Physica A 495 (2018) 69–277. Crossref, Web of Science, Google Scholar
32. L. Xi, Q. Ye and J. Gu , Average geodesic distance of node-weighted Sierpinski networks, Fractals 27(07) (2019) 1950110. Link, Web of Science, Google Scholar
33. J. Deng, Q. Ye and Q. Wang , Weighted average geodesic distance of Vicsek network, Physica A 527 (2019) 121327. Crossref, Web of Science, Google Scholar
34. J. Deng and Q. Wang , Asymptotic formula of average distances on fractal networks modeled by Sierpinski tetrahedron, Fractals 27(7) (2019) 1950120. Link, Web of Science, Google Scholar
35. F. Huang, M. Dai, J. Zhu and W. Su , Scaling of average shortest distance of two colored substitution networks, IEEE Trans. Netw. Sci. Eng. 7(4) (2020) 3067–3073. Crossref, Web of Science, Google Scholar
36. M. Dai, F. Huang, J. Zhu, Y. Guo, Y. Li, H. Qiu and W. Su , Structure properties and weighted average geodesic distances of the Sierpinski carpet fractal networks, Phys. Scr. 95(6) (2020) 065210. Crossref, Web of Science, Google Scholar
37. Y. Liu, M. Dai and Y. Guo , Weighted average geodesic distance of Vicsek network in three-dimensional space, Int. J. Mod. Phys. B 35(5) (2021) 2150077. Link, Web of Science, Google Scholar
38. Y. Liu, M. Dai and Y, Guo , Weight-dependent walks and average shortest weighted path on the weighted iterated friendship graphs, Fractals 30(6) (2022) 2250109. Link, Web of Science, Google Scholar
39. Q. Zhang, Y. Xue, D. Wang and M. Niu , Asymptotic formula on average path length in a hierarchical scale-free network with fractal structure, Chaos Solitons Fractals 122 (2019) 196–201. Crossref, Web of Science, Google Scholar
40. C. Zeng, M. Zhou and Y. Xue , Small-world and scale-free properties of the n-dimensional Sierpinski cube networks, Fractals 28(1) (2020) 2050001. Link, Web of Science, Google Scholar
41. C. Zeng, Y. Xue and M. Zhou , Fractal networks on Sierpinski-type polygon, Fractals 28(5) (2020) 2050087. Link, Web of Science, Google Scholar
42. C. Zeng, Y. Huang and Y. Xue , Average geodesic distance of Sierpinski-type networks, Fractals 30(4) (2022) 2250088. Link, Web of Science, Google Scholar
43. C. Zeng, Y. Huang and Y. Xue , Fractal networks with hierarchical structure: Mean fermat distance and small-world effect, Mod. Phys. Lett. B 36(22) (2022) 2250109. Link, Web of Science, Google Scholar
44. L. Peng, C. Zeng, D. Chen, Y. Xue and Z. Zhao , Average Fermat distance of a pseudo-fractal hierarchical scale-free network, Fractals 31(1) (2023) 2350013. Link, Web of Science, Google Scholar
45. M. Gromov , Hyperbolic groups, in Essays in Group Theory, ed. S. M. Gersten , Math. Sci. Res. Inst. Publ., Vol. 8 (Springer, 1987), pp. 75–263. Crossref, Google Scholar