On <i>D<sub>α</sub></i>-Spectrum of the Weakly Zero-Divisor Graph of ℤ<i><sub>n</sub></i>

On <i>D<sub>α</sub></i>-Spectrum of the Weakly Zero-Divisor Graph of ℤ<i><sub>n</sub></i>

Let us consider the finite commutative ring <i>R</i>, whose unity is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≠</mo><mn>0</mn><mo>...

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Bibliographic Details
Main Authors: Amal S. Alali, Mohd Rashid, Asif Imtiyaz Ahmad Khan, Muzibur Rahman Mozumder
Format: Article
Language:English
Published: MDPI AG 2025-07-01
Series:Mathematics
Subjects:
ring of integers modulo <i>n</i>
weakly zero-divisor graph
spectrum of graph
<i>D<sub>α</sub></i>-matrix
Online Access:https://www.mdpi.com/2227-7390/13/15/2385
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Summary:Let us consider the finite commutative ring <i>R</i>, whose unity is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≠</mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> Its weakly zero-divisor graph, represented 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>R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is a basic undirected graph with two distinct vertices, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</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>c</mi><mn>2</mn></msub></semantics></math></inline-formula>, that 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></mrow></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ann</mi><mo>(</mo><msub><mi>c</mi><mn>1</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><mi>s</mi><mo>∈</mo></mrow></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ann</mi><mo>(</mo><msub><mi>c</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> that satisfy 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>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> be the distance matrix and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> be the diagonal matrix of the vertex transmissions in basic undirected connected graph <i>G</i>. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>α</mi></msub></semantics></math></inline-formula> matrix of graph <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>T</mi><mi>r</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. This article finds the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>α</mi></msub></semantics></math></inline-formula> spectrum for the 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>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> for various values of <i>n</i> and also shows that <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>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><msub><mi>ϑ</mi><mn>1</mn></msub><msub><mi>ϑ</mi><mn>2</mn></msub><msub><mi>ϑ</mi><mn>3</mn></msub><mo>…</mo><msub><mi>ϑ</mi><mi>t</mi></msub><msubsup><mi>η</mi><mn>1</mn><msub><mi>d</mi><mn>1</mn></msub></msubsup><msubsup><mi>η</mi><mn>2</mn><msub><mi>d</mi><mn>2</mn></msub></msubsup><mo>…</mo><msubsup><mi>η</mi><mi>s</mi><msub><mi>d</mi><mi>s</mi></msub></msubsup><mrow><mo>(</mo><msub><mi>d</mi><mi>i</mi></msub><mo>≥</mo><mn>2</mn><mo>,</mo><mi>t</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mi>s</mi><mo>≥</mo><mn>0</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ϑ</mi><mi>i</mi></msub></semantics></math></inline-formula>’s and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>η</mi><mi>i</mi></msub></semantics></math></inline-formula>’s are the distinct primes, is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>α</mi></msub></semantics></math></inline-formula> integral.
ISSN:2227-7390

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