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

Amal S. Alali (), Shahbaz Ali (), Noor Hassan, Ali M. Mahnashi, Yilun Shang and Abdullah Assiry
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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

Mathematics, 2023, vol. 11, issue 18, 1-25

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.

Keywords: algebraic graph theory; algebraic structure graph; commutative ring; zero-divisor graphs; M -polynomials; Zagreb group indices (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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