On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ<sub>n</sub>
For a finite commutative ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">R</mi></semantics></math></inline-formula> with identity <inline-formu...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/20/4310 |
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| author | Nazim Nadeem Ur Rehman Ahmad Alghamdi |
| author_facet | Nazim Nadeem Ur Rehman Ahmad Alghamdi |
| author_sort | Nazim |
| collection | DOAJ |
| description | For a finite commutative ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">R</mi></semantics></math></inline-formula> with identity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>, the weakly zero-divisor graph of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">R</mi></semantics></math></inline-formula> denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi mathvariant="fraktur">R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a simple undirected graph having vertex set as a set of non-zero zero-divisors of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">R</mi></semantics></math></inline-formula> and two distinct vertices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">a</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">b</mi></semantics></math></inline-formula> are adjacent if and only if there exist elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>ann</mi><mo>(</mo><mi mathvariant="fraktur">a</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>ann</mi><mo>(</mo><mi mathvariant="fraktur">b</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfying the condition <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>. The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi mathvariant="fraktur">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Specifically, the investigation is carried out on the weakly zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi mathvariant="fraktur">n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> for various values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">n</mi></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T21:04:41Z |
| format | Article |
| id | doaj.art-604848ab3daa4dc59ba45512df2b9d1a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T21:04:41Z |
| publishDate | 2023-10-01 |
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| series | Mathematics |
| spelling | doaj.art-604848ab3daa4dc59ba45512df2b9d1a2023-11-19T17:14:11ZengMDPI AGMathematics2227-73902023-10-011120431010.3390/math11204310On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ<sub>n</sub>Nazim0Nadeem Ur Rehman1Ahmad Alghamdi2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaMathematics Department, Faculty of Sciences, Umm Al-Qura University, P.O. Box 14035, Makkah 21955, Saudi ArabiaFor a finite commutative ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">R</mi></semantics></math></inline-formula> with identity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>, the weakly zero-divisor graph of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">R</mi></semantics></math></inline-formula> denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi mathvariant="fraktur">R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a simple undirected graph having vertex set as a set of non-zero zero-divisors of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">R</mi></semantics></math></inline-formula> and two distinct vertices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">a</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">b</mi></semantics></math></inline-formula> are adjacent if and only if there exist elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>ann</mi><mo>(</mo><mi mathvariant="fraktur">a</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>ann</mi><mo>(</mo><mi mathvariant="fraktur">b</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfying the condition <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>. The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><mi mathvariant="fraktur">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Specifically, the investigation is carried out on the weakly zero-divisor graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>Γ</mo><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi mathvariant="fraktur">n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> for various values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">n</mi></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/11/20/4310normalized Laplacian spectraweakly zero-divisor graphring of integers modulo nEuler totient function |
| spellingShingle | Nazim Nadeem Ur Rehman Ahmad Alghamdi On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ<sub>n</sub> Mathematics normalized Laplacian spectra weakly zero-divisor graph ring of integers modulo n Euler totient function |
| title | On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ<sub>n</sub> |
| title_full | On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ<sub>n</sub> |
| title_fullStr | On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ<sub>n</sub> |
| title_full_unstemmed | On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ<sub>n</sub> |
| title_short | On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ<sub>n</sub> |
| title_sort | on normalized laplacian spectra of the weakly zero divisor graph of the ring z sub n sub |
| topic | normalized Laplacian spectra weakly zero-divisor graph ring of integers modulo n Euler totient function |
| url | https://www.mdpi.com/2227-7390/11/20/4310 |
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