ECCENTRIC CONNECTIVITY INDEX AND ECCENTRIC DISTANCE SUM OF VICSEK FRACTAL

Yunfeng Xiao, Junhao Peng, Long Gao and Zhenhua Yuan
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Yunfeng Xiao: School of Mathematical and Information Science, Guangzhou University, Guangzhou 510006, P. R. China2Guangdong Provincial Key Laboratory Co-Sponsored by Province and City of Information Security Technology, Guangzhou University, Guangzhou 510006, P. R. China3Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, P. R. China
Junhao Peng: School of Mathematical and Information Science, Guangzhou University, Guangzhou 510006, P. R. China2Guangdong Provincial Key Laboratory Co-Sponsored by Province and City of Information Security Technology, Guangzhou University, Guangzhou 510006, P. R. China3Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, P. R. China
Long Gao: School of Mathematical and Information Science, Guangzhou University, Guangzhou 510006, P. R. China2Guangdong Provincial Key Laboratory Co-Sponsored by Province and City of Information Security Technology, Guangzhou University, Guangzhou 510006, P. R. China3Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, P. R. China
Zhenhua Yuan: School of Mathematical and Information Science, Guangzhou University, Guangzhou 510006, P. R. China2Guangdong Provincial Key Laboratory Co-Sponsored by Province and City of Information Security Technology, Guangzhou University, Guangzhou 510006, P. R. China3Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, P. R. China

FRACTALS (fractals), 2024, vol. 32, issue 06, 1-13

Abstract: The Eccentric Connectivity Index (ECI) and Eccentric Distance Sum (EDS) are two important topological indices with wide applications in biology and chemistry. The Vicsek Fractal (VF) is a notable structure in both polymer physics and network science, and is also known as a regular hyperbranched polymer in the academic realm. In this study, the ECI and EDS of the VF are evaluated analytically by exploring the self-similarity of the structure. First, we derive the recurrence formulas for the ECI and EDS with respect to the generation t of the VF. Then, exact results for the ECI and EDS of the VF are obtained by solving the recurrence relations. The results indicate that the ECI and EDS of the VF scale with a network size of Nt are ECI ∼ Nt1+logp+13 and EDS ∼ Nt2+2logp+13, respectively. The results obtained not only reveal the biological activity and physical properties of the VF but also shed light on the control of the beamwidth in the azimuth plane and the design of the maglev centrifugal blood pump.

Keywords: Vicsek Fractal; Eccentric Connectivity Index; Eccentric Distance Sum (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S0218348X24501135

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