Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula>
Let <i>R</i> be a commutative ring with identity, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi>R</mi><mo>)</mo><...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-10-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/12/10/987 |
| _version_ | 1827721780865794048 |
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| author | Nadeem ur Rehman Amal S. Alali Shabir Ahmad Mir Mohd Nazim |
| author_facet | Nadeem ur Rehman Amal S. Alali Shabir Ahmad Mir Mohd Nazim |
| author_sort | Nadeem ur Rehman |
| collection | DOAJ |
| description | Let <i>R</i> be a commutative ring with identity, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> be the set of zero-divisors of <i>R</i>. The weakly zero-divisor graph of <i>R</i> denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is an undirected (simple) graph with vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>*</mo></msup></mrow></semantics></math></inline-formula>, and two distinct vertices <i>x</i> and <i>y</i> are adjacent, if and only if there exist <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>s</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Importantly, it is worth noting that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> contains the zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">W</mi><mo>Γ</mo><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>p</i>, <i>t</i>, and <i>s</i> are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">W</mi><mo>Γ</mo><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T21:26:49Z |
| format | Article |
| id | doaj.art-cb6ef24ff7e9427e96707ae47e29e9af |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T21:26:49Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-cb6ef24ff7e9427e96707ae47e29e9af2023-11-19T15:38:52ZengMDPI AGAxioms2075-16802023-10-01121098710.3390/axioms12100987Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula>Nadeem ur Rehman0Amal S. Alali1Shabir Ahmad Mir2Mohd Nazim3Department of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaDepartment of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaSchool of Basic and Applied Sciences, Faculty of Science and Technology, JSPM University, Pune 412207, IndiaLet <i>R</i> be a commutative ring with identity, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> be the set of zero-divisors of <i>R</i>. The weakly zero-divisor graph of <i>R</i> denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is an undirected (simple) graph with vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>*</mo></msup></mrow></semantics></math></inline-formula>, and two distinct vertices <i>x</i> and <i>y</i> are adjacent, if and only if there exist <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>s</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Importantly, it is worth noting that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> contains the zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">W</mi><mo>Γ</mo><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>p</i>, <i>t</i>, and <i>s</i> are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">W</mi><mo>Γ</mo><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/12/10/987weakly zero-divisor graphtopological indicescommutative ringsgraph parameters |
| spellingShingle | Nadeem ur Rehman Amal S. Alali Shabir Ahmad Mir Mohd Nazim Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> Axioms weakly zero-divisor graph topological indices commutative rings graph parameters |
| title | Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> |
| title_full | Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> |
| title_fullStr | Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> |
| title_full_unstemmed | Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> |
| title_short | Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>t</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula> |
| title_sort | analysis of the zagreb indices over the weakly zero divisor graph of the ring inline formula math display inline semantics mrow msub mi mathvariant double struck z mi mi p mi msub mo mo msub mi mathvariant double struck z mi mi t mi msub mo mo msub mi mathvariant double struck z mi mi s mi msub mrow semantics math inline formula |
| topic | weakly zero-divisor graph topological indices commutative rings graph parameters |
| url | https://www.mdpi.com/2075-1680/12/10/987 |
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