Algebraic Structure Graphs over the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub></semantics></math></inline-formula>: Exploring Topological Indices and Entropies Using <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-Polynomials
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 struc...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/18/3833 |
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| author | Amal S. Alali Shahbaz Ali Noor Hassan Ali M. Mahnashi Yilun Shang Abdullah Assiry |
| author_facet | Amal S. Alali Shahbaz Ali Noor Hassan Ali M. Mahnashi Yilun Shang Abdullah Assiry |
| author_sort | Amal S. Alali |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>G</mi><mo>^</mo></mover></semantics></math></inline-formula> is fully made up of the zero divisors of the modular ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub></semantics></math></inline-formula>, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (mod<i>n</i>), 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. |
| first_indexed | 2024-03-10T22:29:58Z |
| format | Article |
| id | doaj.art-63b1e81e713f4e038101183d6821be7c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T22:29:58Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-63b1e81e713f4e038101183d6821be7c2023-11-19T11:48:15ZengMDPI AGMathematics2227-73902023-09-011118383310.3390/math11183833Algebraic Structure Graphs over the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub></semantics></math></inline-formula>: Exploring Topological Indices and Entropies Using <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-PolynomialsAmal S. Alali0Shahbaz Ali1Noor Hassan2Ali M. Mahnashi3Yilun Shang4Abdullah Assiry5Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaDepartment of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, PakistanDepartment of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, PakistanDepartment of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi ArabiaDepartment of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UKDepartment of Mathematical Sciences, College of Applied Science, Umm Alqura University, Makkah 21955, Saudi ArabiaThe 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>G</mi><mo>^</mo></mover></semantics></math></inline-formula> is fully made up of the zero divisors of the modular ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub></semantics></math></inline-formula>, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (mod<i>n</i>), 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.https://www.mdpi.com/2227-7390/11/18/3833algebraic graph theoryalgebraic structure graphcommutative ringzero-divisor graphs<named-content content-type="inline"><inline-formula> <mml:math id="mm2000003"> <mml:semantics> <mml:mi mathvariant="double-struck">M</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-polynomialsZagreb group indices |
| spellingShingle | Amal S. Alali Shahbaz Ali Noor Hassan Ali M. Mahnashi Yilun Shang Abdullah Assiry Algebraic Structure Graphs over the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub></semantics></math></inline-formula>: Exploring Topological Indices and Entropies Using <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-Polynomials Mathematics algebraic graph theory algebraic structure graph commutative ring zero-divisor graphs <named-content content-type="inline"><inline-formula> <mml:math id="mm2000003"> <mml:semantics> <mml:mi mathvariant="double-struck">M</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-polynomials Zagreb group indices |
| title | Algebraic Structure Graphs over the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub></semantics></math></inline-formula>: Exploring Topological Indices and Entropies Using <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-Polynomials |
| title_full | Algebraic Structure Graphs over the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub></semantics></math></inline-formula>: Exploring Topological Indices and Entropies Using <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-Polynomials |
| title_fullStr | Algebraic Structure Graphs over the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub></semantics></math></inline-formula>: Exploring Topological Indices and Entropies Using <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-Polynomials |
| title_full_unstemmed | Algebraic Structure Graphs over the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub></semantics></math></inline-formula>: Exploring Topological Indices and Entropies Using <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-Polynomials |
| title_short | Algebraic Structure Graphs over the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub></semantics></math></inline-formula>: Exploring Topological Indices and Entropies Using <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-Polynomials |
| title_sort | algebraic structure graphs over the commutative ring inline formula math display inline semantics msub mi mathvariant double struck z mi mi m mi msub semantics math inline formula exploring topological indices and entropies using inline formula math display inline semantics mi mathvariant double struck m mi semantics math inline formula polynomials |
| topic | algebraic graph theory algebraic structure graph commutative ring zero-divisor graphs <named-content content-type="inline"><inline-formula> <mml:math id="mm2000003"> <mml:semantics> <mml:mi mathvariant="double-struck">M</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-polynomials Zagreb group indices |
| url | https://www.mdpi.com/2227-7390/11/18/3833 |
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