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Algebraic Structure Graphs over the Commutative Ring Z m : Exploring Topological Indices and Entropies Using M -Polynomials

Author

Listed:
  • Amal S. Alali

    (Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia)

  • Shahbaz Ali

    (Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan)

  • Noor Hassan

    (Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan)

  • Ali M. Mahnashi

    (Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia)

  • Yilun Shang

    (Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK)

  • Abdullah Assiry

    (Department of Mathematical Sciences, College of Applied Science, Umm Alqura University, Makkah 21955, Saudi Arabia)

Abstract

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph G ^ is fully made up of the zero divisors of the modular ring Z n , the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (mod n ), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

Suggested Citation

  • Amal S. Alali & Shahbaz Ali & Noor Hassan & Ali M. Mahnashi & Yilun Shang & Abdullah Assiry, 2023. "Algebraic Structure Graphs over the Commutative Ring Z m : Exploring Topological Indices and Entropies Using M -Polynomials," Mathematics, MDPI, vol. 11(18), pages 1-25, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3833-:d:1234745
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    References listed on IDEAS

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    1. Chun-Xiao Nie & Fu-Tie Song, 2021. "Entropy of Graphs in Financial Markets," Computational Economics, Springer;Society for Computational Economics, vol. 57(4), pages 1149-1166, April.
    2. Huang, Wencheng & Zhang, Yue & Yu, Yaocheng & Xu, Yifei & Xu, Minhao & Zhang, Rui & De Dieu, Gatesi Jean & Yin, Dezhi & Liu, Zhanru, 2021. "Historical data-driven risk assessment of railway dangerous goods transportation system: Comparisons between Entropy Weight Method and Scatter Degree Method," Reliability Engineering and System Safety, Elsevier, vol. 205(C).
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