On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mstyle mathvariant="bold"><mi mathvariant="double-struck">Z</mi></mstyle><mrow><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">1</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">1</mn></msub></msubsup><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">2</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">2</mn></msub></msubsup></mrow></msub></semantics></math></inline-formula>
The zero-divisor graph of a 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 a nonzero...
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| author | Ali Al Khabyah Nazim Nadeem Ur Rehman |
| author_facet | Ali Al Khabyah Nazim Nadeem Ur Rehman |
| author_sort | Ali Al Khabyah |
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| description | The zero-divisor graph of a 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 a nonzero identity, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><mi mathvariant="fraktur">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is an undirected graph where the 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 mathvariant="fraktur">R</mi><mo>)</mo></mrow><mo>*</mo></msup></mrow></semantics></math></inline-formula> consists of all nonzero 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>. Two distinct vertices <i>a</i> and <i>b</i> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><mi mathvariant="fraktur">R</mi><mo>)</mo></mrow></semantics></math></inline-formula> are adjacent if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>b</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The normalized Laplacian spectrum of zero-divisor graphs has been studied extensively due to its algebraic and combinatorial significance. Notably, Pirzada and his co-authors computed the normalized Laplacian spectrum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><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 specific 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> in the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>p</mi><mi>q</mi><mo>,</mo><msup><mi>p</mi><mn>2</mn></msup><mi>q</mi><mo>,</mo><msup><mi>p</mi><mn>3</mn></msup><mo>,</mo><msup><mi>p</mi><mn>4</mn></msup><mo>}</mo></mrow></semantics></math></inline-formula>, where <i>p</i> and <i>q</i> are distinct primes satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo><</mo><mi>q</mi></mrow></semantics></math></inline-formula>. Motivated by their work, this article investigates the normalized Laplacian spectrum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><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 a more general class of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">n</mi></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">n</mi></semantics></math></inline-formula> is represented as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>p</mi><mn>1</mn><msub><mi>T</mi><mn>1</mn></msub></msubsup><msubsup><mi>p</mi><mn>2</mn><msub><mi>T</mi><mn>2</mn></msub></msubsup></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mn>2</mn></msub></semantics></math></inline-formula> being distinct primes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo><</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>T</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> are positive integers. |
| format | Article |
| id | doaj-art-1e63fda6417244c3881b0c373fc98d0e |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-1e63fda6417244c3881b0c373fc98d0e2025-01-24T13:22:13ZengMDPI AGAxioms2075-16802025-01-011413710.3390/axioms14010037On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mstyle mathvariant="bold"><mi mathvariant="double-struck">Z</mi></mstyle><mrow><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">1</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">1</mn></msub></msubsup><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">2</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">2</mn></msub></msubsup></mrow></msub></semantics></math></inline-formula>Ali Al Khabyah0Nazim1Nadeem Ur Rehman2Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi ArabiaDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaThe zero-divisor graph of a 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 a nonzero identity, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><mi mathvariant="fraktur">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is an undirected graph where the 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 mathvariant="fraktur">R</mi><mo>)</mo></mrow><mo>*</mo></msup></mrow></semantics></math></inline-formula> consists of all nonzero 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>. Two distinct vertices <i>a</i> and <i>b</i> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Γ</mo><mo>(</mo><mi mathvariant="fraktur">R</mi><mo>)</mo></mrow></semantics></math></inline-formula> are adjacent if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>b</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The normalized Laplacian spectrum of zero-divisor graphs has been studied extensively due to its algebraic and combinatorial significance. Notably, Pirzada and his co-authors computed the normalized Laplacian spectrum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><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 specific 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> in the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>p</mi><mi>q</mi><mo>,</mo><msup><mi>p</mi><mn>2</mn></msup><mi>q</mi><mo>,</mo><msup><mi>p</mi><mn>3</mn></msup><mo>,</mo><msup><mi>p</mi><mn>4</mn></msup><mo>}</mo></mrow></semantics></math></inline-formula>, where <i>p</i> and <i>q</i> are distinct primes satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo><</mo><mi>q</mi></mrow></semantics></math></inline-formula>. Motivated by their work, this article investigates the normalized Laplacian spectrum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><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 a more general class of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">n</mi></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">n</mi></semantics></math></inline-formula> is represented as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>p</mi><mn>1</mn><msub><mi>T</mi><mn>1</mn></msub></msubsup><msubsup><mi>p</mi><mn>2</mn><msub><mi>T</mi><mn>2</mn></msub></msubsup></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mn>2</mn></msub></semantics></math></inline-formula> being distinct primes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo><</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>T</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> are positive integers.https://www.mdpi.com/2075-1680/14/1/37normalized Laplacian matrixzero-divisor graphnormalized Laplacian spectrumfinite commutative ring |
| spellingShingle | Ali Al Khabyah Nazim Nadeem Ur Rehman On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mstyle mathvariant="bold"><mi mathvariant="double-struck">Z</mi></mstyle><mrow><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">1</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">1</mn></msub></msubsup><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">2</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">2</mn></msub></msubsup></mrow></msub></semantics></math></inline-formula> Axioms normalized Laplacian matrix zero-divisor graph normalized Laplacian spectrum finite commutative ring |
| title | On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mstyle mathvariant="bold"><mi mathvariant="double-struck">Z</mi></mstyle><mrow><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">1</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">1</mn></msub></msubsup><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">2</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">2</mn></msub></msubsup></mrow></msub></semantics></math></inline-formula> |
| title_full | On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mstyle mathvariant="bold"><mi mathvariant="double-struck">Z</mi></mstyle><mrow><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">1</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">1</mn></msub></msubsup><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">2</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">2</mn></msub></msubsup></mrow></msub></semantics></math></inline-formula> |
| title_fullStr | On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mstyle mathvariant="bold"><mi mathvariant="double-struck">Z</mi></mstyle><mrow><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">1</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">1</mn></msub></msubsup><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">2</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">2</mn></msub></msubsup></mrow></msub></semantics></math></inline-formula> |
| title_full_unstemmed | On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mstyle mathvariant="bold"><mi mathvariant="double-struck">Z</mi></mstyle><mrow><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">1</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">1</mn></msub></msubsup><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">2</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">2</mn></msub></msubsup></mrow></msub></semantics></math></inline-formula> |
| title_short | On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring <inline-formula><math display="inline"><semantics><msub><mstyle mathvariant="bold"><mi mathvariant="double-struck">Z</mi></mstyle><mrow><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">1</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">1</mn></msub></msubsup><msubsup><mi mathvariant="bold-italic">p</mi><mn mathvariant="bold">2</mn><msub><mi mathvariant="bold-italic">T</mi><mn mathvariant="bold">2</mn></msub></msubsup></mrow></msub></semantics></math></inline-formula> |
| title_sort | on the normalized laplacian spectrum of the zero divisor graph of the commutative ring inline formula math display inline semantics msub mstyle mathvariant bold mi mathvariant double struck z mi mstyle mrow msubsup mi mathvariant bold italic p mi mn mathvariant bold 1 mn msub mi mathvariant bold italic t mi mn mathvariant bold 1 mn msub msubsup msubsup mi mathvariant bold italic p mi mn mathvariant bold 2 mn msub mi mathvariant bold italic t mi mn mathvariant bold 2 mn msub msubsup mrow msub semantics math inline formula |
| topic | normalized Laplacian matrix zero-divisor graph normalized Laplacian spectrum finite commutative ring |
| url | https://www.mdpi.com/2075-1680/14/1/37 |
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