Multiplicative Sombor index of trees

Nasrin Dehgardi, Zhibin Du and Yilun Shang
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 2, Pages 453–460
DOI: 10.7546/nntdm.2024.30.2.453-460
Full paper (PDF, 196 Kb)

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Authors and affiliations

Nasrin Dehgardi
Department of Mathematics and Computer Science, Sirjan University of Technology
Sirjan, Iran

Zhibin Du
School of Software, South China Normal University
Foshan, Guangdong 528225, China

Yilun Shang
Department of Computer and Information Sciences, Northumbria University
Newcastle NE1 8ST, United Kingdom

Abstract

For a graph \Omega, the multiplicative Sombor index is defined as

    \[\prod_{SO}(\Omega)=\prod_{ab\in \mathcal{E}(\Omega)}\sqrt{d^2_\Omega(a)+d^2_\Omega(b)},\]

where d_\Omega(a) is the degree of vertex a. Liu [Liu, H. (2022). Discrete Mathematics Letters, 9, 80–85] showed that, when \mathcal{T} is a tree of order n, \prod_{SO}(\mathcal{T})\geqslant \prod_{SO}(P_n)=5(\sqrt{8})^{n-3}. We improved this result and show that, if \mathcal{T} is a tree of order n with maximum degree \cal{D}, then

    \[\prod_{SO}(\mathcal{T})\geqslant \left\{\begin{array}{ll} (5({\cal{D}}^2+4))^{\frac{\cal{D}}{2}}8^{\frac{n-2{\cal{D}}-1}{2}} & {\rm if}\;{\cal{D}}\leqslant\frac{n-1}{2},\\[2mm] ({\cal{D}}^2+1)^{\frac{2{\cal{D}}+1-n}{2}}(5({\cal{D}}^2+4))^{\frac{n-{\cal{D}}-1}{2}} & {\rm if}\;{\cal{D}}>\frac{n-1}{2}. \end{array}\right.\]

Also, we show that equality holds if and only if \mathcal{T} is a spider whose all legs have length less than three or all legs have length more than one.

Keywords

  • Sombor index
  • Multiplicative Sombor index
  • Trees

2020 Mathematics Subject Classification

  • 05C07

References

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Manuscript history

  • Received: 17 October 2023
  • Revised: 3 June 2024
  • Accepted: 18 July 2024
  • Online First: 20 July 2024

Copyright information

Ⓒ 2024 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Dehgardi, N., Du, Z., & Shang, Y. (2024). Multiplicative Sombor index of trees. Notes on Number Theory and Discrete Mathematics, 30(2), 453-460, DOI: 10.7546/nntdm.2024.30.2.453-460.

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