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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/37 |
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| Summary: | 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. |
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| ISSN: | 2075-1680 |